# Geometry

## Polygons, Perimeter, and Circumference

### Learning Objectives

After completing this section, you should be able to:
1. Identify polygons by their sides.
2. Identify polygons by their characteristics.
3. Calculate the perimeter of a polygon.
4. Calculate the sum of the measures of a polygon’s interior angles.
5. Calculate the sum of the measures of a polygon’s exterior angles.
6. Calculate the circumference of a circle.
7. Solve application problems involving perimeter and circumference.
In our homes, on the road, everywhere we go, polygonal shapes are so common that we cannot count the many uses. Traffic signs, furniture, lighting, clocks, books, computers, phones, and so on, the list is endless. Many applications of polygonal shapes are for practical use, because the shapes chosen are the best for the purpose.
Modern geometric patterns in fabric design have become more popular with time, and they are used for the beauty they lend to the material, the window coverings, the dresses, or the upholstery. This art is not done for any practical reason, but only for the interest these shapes can create, for the pure aesthetics of design.
When designing fabrics, one has to consider the perimeter of the shapes, the triangles, the hexagons, and all polygons used in the pattern, including the circumference of any circular shapes. Additionally, it is the relationship of one object to another and experimenting with different shapes, changing perimeters, or changing angle measurements that we find the best overall design for the intended use of the fabric. In this section, we will explore these properties of polygons, the perimeter, the calculation of interior and exterior angles of polygons, and the circumference of a circle.

### Identifying Polygons

A polygon is a closed, two-dimensional shape classified by the number of straight-line sides. See  for some examples. We show only up to eight-sided polygons, but there are many, many more.
If all the sides of a polygon have equal lengths and all the angles are equal, they are called regular polygons. However, any shape with sides that are line segments can classify as a polygon. For example, the first two shapes, shown in  and , are both pentagons because they each have five sides and five vertices. The third shape  is a hexagon because it has six sides and six vertices. We should note here that the hexagon in  is a concave hexagon, as opposed to the first two shapes, which are convex pentagons. Technically, what makes a polygon concave is having an interior angle that measures greater than . They are hollowed out, or cave in, so to speak. Convex refers to the opposite effect where the shape is rounded out or pushed out.
While there are variations of all polygons, quadrilaterals contain an additional set of figures classified by angles and whether there are one or more pairs of parallel sides. See .

### Perimeter

Perimeter refers to the outside measurements of some area or region given in linear units. For example, to find out how much fencing you would need to enclose your backyard, you will need the perimeter. The general definition of perimeter is the sum of the lengths of the sides of an enclosed region. For some geometric shapes, such as rectangles and circles, we have formulas. For other shapes, it is a matter of just adding up the side lengths.
A rectangle is defined as part of the group known as quadrilaterals, or shapes with four sides. A rectangle has two sets of parallel sides with four angles. To find the perimeter of a rectangle, we use the following formula:
For example, to find the length of a rectangle that has a perimeter of 24 inches and a width of 4 inches, we use the formula. Thus,

The length is 8 units.
The perimeter of a regular polygon with  sides is given as . For example, the perimeter of an equilateral triangle, a triangle with three equal sides, and a side length of 7 cm is .

### Sum of Interior and Exterior Angles

To find the sum of the measurements of interior angles of a regular polygon, we have the following formula.
For example, if we want to find the sum of the interior angles in a parallelogram, we have

Similarly, to find the sum of the interior angles inside a regular heptagon, we have

To find the measure of each interior angle of a regular polygon with  sides, we have the following formula.
For example, find the measure of an interior angle of a regular heptagon, as shown in . We have

An exterior angle of a regular polygon is an angle formed by extending a side length beyond the closed figure. The measure of an exterior angle of a regular polygon with  sides is found using the following formula:
In , we have a regular hexagon . By extending the lines of each side, an angle is formed on the exterior of the hexagon at each vertex. The measure of each exterior angle is found using the formula, .
Now, an important point is that the sum of the exterior angles of a regular polygon with  sides equals  This implies that when we multiply the measure of one exterior angle by the number of sides of the regular polygon, we should get  For the example in , we multiply the measure of each exterior angle, , by the number of sides, six. Thus, the sum of the exterior angles is

### Circles and Circumference

The perimeter of a circle is called the circumference. To find the circumference, we use the formula  where  is the diameter, the distance across the center, or  where  is the radius.
The radius is ½ of the diameter of a circle. The symbol  is the ratio of the circumference to the diameter. Because this ratio is constant, our formula is accurate for any size circle. See .
Let the radius be equal to 3.5 inches. Then, the circumference is

### Check Your Understanding

### Key Terms

1. perimeter
2. polygon
3. pentagon
4. hexagon
5. heptagon
6. octagon
7. quadrilateral
8. trapezoid
9. parallelogram
10. circumference

### Key Concepts

1. Regular polygons are closed, two-dimensional figures that have equal side lengths. They are named for the number of their sides.
2. The perimeter of a polygon is the measure of the outline of the shape. We determine a shape’s perimeter by calculating the sum of the lengths of its sides.
3. The sum of the interior angles of a regular polygon with  sides is found using the formula  The measure of a single interior angle of a regular polygon with  sides is determined using the formula
4. The sum of the exterior angles of a regular polygon is  The measure of a single exterior angle of a regular polygon with  sides is found using the formula
5. The circumference of a circle is  where  is the radius, or  and  is the diameter.

### Formulas

The formula for the perimeter  of a rectangle is , twice the length  plus twice the width .
The sum of the interior angles of a polygon with  sides is given by

The measure of each interior angle of a regular polygon with  sides is given by

To find the measure of an exterior angle of a regular polygon with  sides we use the formula

The circumference of a circle is found using the formula  where  is the diameter of the circle, or  where  is the radius.


# Geometry

## Tessellations

### Learning Objectives

After completing this section, you should be able to:
1. Apply translations, rotations, and reflections.
2. Determine if a shape tessellates.
The illustration shown above () is an unusual pattern called a Penrose tiling. Notice that there are two types of shapes used throughout the pattern: smaller green parallelograms and larger blue parallelograms. What's interesting about this design is that although it uses only two shapes over and over, there is no repeating pattern.
In this section, we will focus on patterns that do repeat. Repeated patterns are found in architecture, fabric, floor tiles, wall patterns, rug patterns, and many unexpected places as well. It may be a simple hexagon-shaped floor tile, or a complex pattern composed of several different motifs. These two-dimensional designs are called regular (or periodic) tessellations. There are countless designs that may be classified as regular tessellations, and they all have one thing in common—their patterns repeat and cover the plane.
We will explore how tessellations are created and experiment with making some of our own as well. The topic of tessellations belongs to a field in mathematics called transformational geometry, which is a study of the ways objects can be moved while retaining the same shape and size. These movements are termed rigid motions and symmetries.

### Tessellation Properties and Transformations

A regular tessellation means that the pattern is made up of congruent regular polygons, same size and shape, including some type of movement; that is, some type of transformation or symmetry. Here we consider the rigid motions of translations, rotations, reflections, or glide reflections. A plane of tessellations has the following properties:

In , the tessellation is made up of squares. There are four squares meeting at a vertex. An interior angle of a square is  and the sum of four interior angles is  In , the tessellation is made up of regular hexagons. There are three hexagons meeting at each vertex. The interior angle of a hexagon is  and the sum of three interior angles is  Both tessellations will fill the plane, there are no gaps, the sum of the interior angle meeting at the vertex is  and both are achieved by translation transformations. These tessellations work because all the properties of a tessellation are present.
The movements or rigid motions of the shapes that define tessellations are classified as translations, rotations, reflections, or glide reflections. Let’s first define these movements and then look at some examples showing how these transformations are revealed.

### Translation

A translation is a movement that shifts the shape vertically, horizontally, or on the diagonal. Consider the trapezoid  in . We have translated it 3 units to the right and 3 units up. That means every corner is moved by the number of units and in the direction specified. Mathematicians will indicate this movement with a vector, an arrow that is drawn to illustrate the criteria and the magnitude of the translation. The location of the translated trapezoid is marked with the vertices,  but it is still the exact same shape and size as the original trapezoid .

### Rotation

The rotation transformation occurs when you rotate a shape about a point and at a predetermined angle. In , the triangle is rotated around the rotation point by  and then translated 7 units up and 4 units over to the right. That means that each corner is translated to the new location by the same number of units and in the same direction.
We can see that  is mapped to  by a rotation of  up and to the right. If rotated again by , the triangle would be upside down.

### Reflection

A reflection is the third transformation. A shape is reflected about a line and the new shape becomes a mirror image. You can reflect the shape vertically, horizontally, or on the diagonal. There are two shapes in . The quadrilateral is reflected horizontally; the arrow shape is reflected vertically.

### Glide Reflection

The glide reflection is the fourth transformation. It is a combination of a reflection and a translation. This can occur by first reflecting the shape and then gliding or translating it to its new location, or by translating first and then reflecting. The example in  shows a trapezoid, which is reflected over the dashed line, so it appears upside down. Then, we shifted the shape horizontally by 6 units to the right. Whether we use the glide first or the reflection first, the end result is the same in most cases. However, the tessellation shown in the next example can only be achieved by a reflection first and then a translation.

### Interior Angles

The sum of the interior angles of a tessellation is . In , the tessellation is made of six triangles formed into the shape of a hexagon. Each angle inside a triangle equals , and the six vertices meet the sum of those interior angles, .
In , the tessellation is made up of trapezoids, such that two of the interior angles of each trapezoid equals  and the other two angles equal . Thus, the sum of the interior angles where the vertices of four trapezoids meet equals .
These tessellations illustrate the property that the shapes meet at a vertex where the interior angles sum to .

### Tessellating Shapes

We might think that all regular polygons will tessellate the plane by themselves. We have seen that squares do and hexagons do. The pattern of squares in  is a translation of the shape horizontally and vertically. The hexagonal pattern in , is translated horizontally, and then on the diagonal, either to the right or to the left. This particular pattern can also be formed by rotations. Both tessellations are made up of congruent shapes and each shape fits in perfectly as the pattern repeats.
We have also seen that equilateral triangles will tessellate the plane without gaps or overlaps, as shown in . The pattern is made by a reflection and a translation. The darker side is the face of the triangle and the lighter side is the back of the triangle, shown by the reflection. Each triangle is reflected and then translated on the diagonal.
Escher experimented with all regular polygons and found that only the ones mentioned, the equilateral triangle, the square, and the hexagon, will tessellate the plane by themselves. Let’s try a few other regular polygons to observe what Escher found.
Just because regular pentagons do not tessellate the plane by themselves does not mean that there are no pentagons that tessellate the plane, as we see in .
Another example of an irregular polygon that tessellates the plane is by using the obtuse irregular triangle from a previous example. What transformations should be performed to produce the tessellation shown in ?
First, the triangle is reflected over the tip at point , and then translated to the right and joined with the original triangle to form a parallelogram. The parallelogram is then translated on the diagonal and to the right and to the left.

### Naming

A tessellation of squares is named by choosing a vertex and then counting the number of sides of each shape touching the vertex. Each square in the tessellation shown in  has four sides, so starting with square , the first number is 4, moving around counterclockwise to the next square meeting the vertex, square , we have another 4, square  adds another 4, and finally square  adds a fourth 4. So, we would name this tessellation a 4.4.4.4.
The hexagon tessellation, shown in  has six sides to the shape and three hexagons meet at the vertex. Thus, we would name this a 6.6.6. The triangle tessellation, shown in  has six triangles meeting the vertex. Each triangle has three sides. Thus, we name this a 3.3.3.3.3.3.

### Check Your Understanding

### Key Terms

1. tessellation
2. translation
3. reflection
4. rotation
5. glide reflection

### Key Concepts

1. A tessellation is a particular pattern composed of shapes, usually polygons, that repeat and cover the plane with no gaps or overlaps.
2. Properties of tessellations include rigid motions of the shapes called transformations. Transformations refer to translations, rotations, reflections, and glide reflections. Shapes are transformed in such a way to create a pattern.

### Videos

1. M.C. Escher: How to Create a Tessellation
2. The Mathematical Art of M.C. Escher


# Geometry

## Area

### Learning Objectives

After completing this section, you should be able to:
1. Calculate the area of triangles.
2. Calculate the area of quadrilaterals.
3. Calculate the area of other polygons.
4. Calculate the area of circles.
Some areas carry more importance than other areas. Did you know that in a baseball game, when the player hits the ball and runs to first base that he must run within a 6-foot wide path? If he veers off slightly to the right, he is out. In other words, a few inches can be the difference in winning or losing a game. Another example is real estate. On Manhattan Island, one square foot of real estate is worth far more than real estate in practically any other area of the country. In other words, we place a value on area. As the context changes, so does the value.
Area refers to a region measured in square units, like a square mile or a square foot. For example, to purchase tile for a kitchen floor, you would need to know how many square feet are needed because tile is sold by the square foot. Carpeting is sold by the square yard. As opposed to linear measurements like perimeter, which in in linear units. For example, fencing is sold in linear units, a linear foot or yard. Linear dimensions refer to an outline or a boundary. Square units refer to the area within that boundary. Different items may have different units, but either way, you must know the linear dimensions to calculate the area.
Many geometric shapes have formulas for calculating areas, such as triangles, regular polygons, and circles. To calculate areas for many irregular curves or shapes, we need calculus. However, in this section, we will only look at geometric shapes that have known area formulas. The notation for area, as mentioned, is in square units and we write sq in or sq cm, or use an exponent, such as  or  Note that linear measurements have no exponent above the units or we can say that the exponent is 1.

### Area of Triangles

The formula for the area of a triangle is given as follows.
For example, consider the triangle in .
The base measures 4 cm and the height measures 5 cm. Using the formula, we can calculate the area:

In , the triangle has a base equal to 7 cm and a height equal to 3.5 cm. Notice that we can only find the height by dropping a perpendicular to the base. The area is then

### Area of Quadrilaterals

A quadrilateral is a four-sided polygon with four vertices. Some quadrilaterals have either one or two sets of parallel sides. The set of quadrilaterals include the square, the rectangle, the parallelogram, the trapezoid, and the rhombus. The most common quadrilaterals are the square and the rectangle.

### Square

In , a  grid is represented with twelve  squares across each row, and twelve  squares down each column. If you count the little squares, the sum equals 144 squares. Of course, you do not have to count little squares to find area—we have a formula. Thus, the formula for the area of a square, where , is . The area of the square in  is

### Rectangle

Similarly, the area for a rectangle is found by multiplying length times width. The rectangle in  has width equal to 5 in and length equal to 12 in. The area is
Many everyday applications require the use of the perimeter and area formulas. Suppose you are remodeling your home and you want to replace all the flooring. You need to know how to calculate the area of the floor to purchase the correct amount of tile, or hardwood, or carpet. If you want to paint the rooms, you need to calculate the area of the walls and the ceiling to know how much paint to buy, and the list goes on. Can you think of other situations where you might need to calculate area?

### Parallelogram

The area of a parallelogram can be found using the formula for the area of a triangle. Notice in , if we cut a diagonal across the parallelogram from one vertex to the opposite vertex, we have two triangles. If we multiply the area of a triangle by 2, we have the area of a parallelogram:

For example, if we have a parallelogram with the base be equal to 10 inches and the height equal to 5 inches, the area will be

### Trapezoid

Another quadrilateral is the trapezoid. A trapezoid has one set of parallel sides or bases. The formula for the area of a trapezoid with parallel bases  and  and height  is given here.
For example, find the area of the trapezoid in  that has base  equal to 8 cm, base  equal to 6 cm, and height equal to 6 cm.
The area is .

### Rhombus

The rhombus has two sets of parallel sides. To find the area of a rhombus, there are two formulas we can use. One involves determining the measurement of the diagonals.
For our purposes here, we will use the formula that uses diagonals. For example, if the area of a rhombus is  and the measure of  find the measure of  To solve this problem, we input the known values into the formula and solve for the unknown. See .
We have that

### Area of Polygons

To find the area of a regular polygon, we need to learn about a few more elements. First, the apothem  of a regular polygon is a line segment that starts at the center and is perpendicular to a side. The radius  of a regular polygon is also a line segment that starts at the center but extends to a vertex. See .
For example, consider the regular hexagon shown in  with a side length of 4 cm, and the apothem measures
We have the perimeter,  We have the apothem as  Then, the area is:

### Changing Units

Often, we have the need to change the units of one or more items in a problem to find a solution. For example, suppose you are purchasing new carpet for a room measured in feet, but carpeting is sold in terms of yards. You will have to convert feet to yards to purchase the correct amount of carpeting. Or, you may need to convert centimeters to inches, or feet to meters. In each case, it is essential to use the correct equivalency.

### Area of Circles

Just as the circumference of a circle includes the number  so does the formula for the area of a circle. Recall that  is a non-terminating, non-repeating decimal number: . It represents the ratio of the circumference to the diameter, so it is a critical number in the calculation of circumference and area.
For example, to find the of the circle with radius equal to 3 cm, as shown in , is found using the formula
We have

### Area within Area

Suppose you want to install a round hot tub on your backyard patio. How would you calculate the space needed for the hot tub? Or, let’s say that you want to purchase a new dining room table, but you are not sure if you have enough space for it. These are common issues people face every day. So, let’s take a look at how we solve these problems.

### Check Your Understanding

### Key Terms

1. triangle
2. square
3. rectangle
4. rhombus
5. apothem
6. radius
7. circle

### Key Concepts

1. The area  of a triangle is found with the formula  where  is the base and  is the height.
2. The area of a parallelogram is found using the formula  where  is the base and  is the height.
3. The area of a rectangle is found using the formula  where  is the length and  is the width.
4. The area of a trapezoid is found using the formula  where  is the height,  is the length of one base, and  is the length of the other base.
5. The area of a rhombus is found using the formula  where  is the length of one diagonal and  is the length of the other diagonal.
6. The area of a regular polygon is found using the formula  where  is the apothem and  is the perimeter.
7. The area of a circle is found using the formula  where  is the radius.

### Formulas

The area of a triangle is given as  where  represents the base and  represents the height.
The formula for the area of a square is  or
The area of a rectangle is given as
The area of a parallelogram is
The formula for the area of a trapezoid is given as
The area of a rhombus is found using one of these formulas:

The area of a regular polygon is found with the formula  where  is the apothem and  is the perimeter.
The area of a circle is given as  where  is the radius.


# Geometry

## Volume and Surface Area

### Learning Objectives

After completing this section, you should be able to:
1. Calculate the surface area of right prisms and cylinders.
2. Calculate the volume of right prisms and cylinders.
3. Solve application problems involving surface area and volume.
Volume and surface area are two measurements that are part of our daily lives. We use volume every day, even though we do not focus on it. When you purchase groceries, volume is the key to pricing. Judging how much paint to buy or how many square feet of siding to purchase is based on surface area. The list goes on. An example is a three-dimensional rendering of a floor plan. These types of drawings make building layouts far easier to understand for the client. It allows the viewer a realistic idea of the product at completion; you can see the natural space, the volume of the rooms. This section gives you practical information you will use consistently. You may not remember every formula, but you will remember the concepts, and you will know where to go should you want to calculate volume or surface area in the future.
We will concentrate on a few particular types of three-dimensional objects: right prisms and right cylinders. The adjective “right” refers to objects such that the sides form a right angle with the base. We will look at right rectangular prisms, right triangular prisms, right hexagonal prisms, right octagonal prisms, and right cylinders. Although, the principles learned here apply to all right prisms.

### Three-Dimensional Objects

In geometry, three-dimensional objects are called geometric solids. Surface area refers to the flat surfaces that surround the solid and is measured in square units. Volume refers to the space inside the solid and is measured in cubic units. Imagine that you have a square flat surface with width and length. Adding the third dimension adds depth or height, depending on your viewpoint, and now you have a box. One way to view this concept is in the Cartesian coordinate three-dimensional space. The -axis and the -axis are, as you would expect, two dimensions and suitable for plotting two-dimensional graphs and shapes. Adding the -axis, which shoots through the origin perpendicular to the -plane, and we have a third dimension. See .
Here is another view taking the two-dimensional square to a third dimension. See .
To study objects in three dimensions, we need to consider the formulas for surface area and volume. For example, suppose you have a box () with a hinged lid that you want to use for keeping photos, and you want to cover the box with a decorative paper. You would need to find the surface area to calculate how much paper is needed. Suppose you need to know how much water will fill your swimming pool. In that case, you would need to calculate the volume of the pool. These are just a couple of examples of why these concepts should be understood, and familiarity with the formulas will allow you to make use of these ideas as related to right prisms and right cylinders.

### Right Prisms

A right prism is a particular type of three-dimensional object. It has a polygon-shaped base and a congruent, regular polygon-shaped top, which are connected by the height of its lateral sides, as shown in . The lateral sides form a right angle with the base and the top. There are rectangular prisms, hexagonal prisms, octagonal prisms, triangular prisms, and so on.
Generally, to calculate surface area, we find the area of each side of the object and add the areas together. To calculate volume, we calculate the space inside the solid bounded by its sides.
In , we have three views of a right hexagonal prism. The regular hexagon is the base and top, and the lateral faces are the rectangular regions perpendicular to the base. We call it a right prism because the angle formed by the lateral sides to the base is  See .
The first image is a view of the figure straight on with no rotation in any direction. The middle figure is the base or the top. The last figure shows you the solid in three dimensions. To calculate the surface area of the right prism shown in , we first determine the area of the hexagonal base and multiply that by 2, and then add the perimeter of the base times the height. Recall the area of a regular polygon is given as  where  is the apothem and  is the perimeter. We have that

Then, the surface area of the hexagonal prism is

To find the volume of the right hexagonal prism, we multiply the area of the base by the height using the formula  The base is  and the height is 20 . Thus,

### Right Cylinders

There are similarities between a prism and a cylinder. While a prism has parallel congruent polygons as the top and the base, a right cylinder is a three-dimensional object with congruent circles as the top and the base. The lateral sides of a right prism make a  angle with the polygonal base, and the side of a cylinder, which unwraps as a rectangle, makes a  angle with the circular base.
Right cylinders are very common in everyday life. Think about soup cans, juice cans, soft drink cans, pipes, air hoses, and the list goes on.
In , imagine that the cylinder is cut down the 12-inch side and rolled out. We can see that the cylinder side when flat forms a rectangle. The  formula includes the area of the circular base, the circular top, and the area of the rectangular side. The length of the rectangular side is the circumference of the circular base. Thus, we have the formula for total surface area of a right cylinder.
To find the volume of the cylinder, we multiply the area of the base with the height.

### Applications of Surface Area and Volume

The following are just a small handful of the types of applications in which surface area and volume are critical factors. Give this a little thought and you will realize many more practical uses for these procedures.

### Optimization

Problems that involve optimization are ones that look for the best solution to a situation under some given conditions. Generally, one looks to calculus to solve these problems. However, many geometric applications can be solved with the tools learned in this section. Suppose you want to make some throw pillows for your sofa, but you have a limited amount of fabric. You want to make the largest pillows you can from the fabric you have, so you would need to figure out the dimensions of the pillows that will fit these criteria. Another situation might be that you want to fence off an area in your backyard for a garden. You have a limited amount of fencing available, but you would like the garden to be as large as possible. How would you determine the shape and size of the garden? Perhaps you are looking for maximum volume or minimum surface area. Minimum cost is also a popular application of optimization. Let’s explore a few examples.

### Check Your Understanding

### Key Terms

1. surface area
2. volume
3. right prism
4. right cylinder

### Key Concepts

1. A right prism is a three-dimensional object that has a regular polygonal face and congruent base such that that lateral sides form a  angle with the base and top. The surface area  of a right prism is found using the formula  where  is the area of the base,  is the perimeter of the base, and  is the height. The volume  of a right prism is found using the formula  where  is the area of the base and  is the height.
2. A right cylinder is a three-dimensional object with a circle as the top and a congruent circle is the base, and the side forms a  angle to the base and top. The surface area of a right cylinder is found using the formula  where  is the radius and  is the height. The volume is found using the formula  where  is the radius and  is the height.

### Formulas

The formula for the surface area of a right prism is equal to twice the area of the base plus the perimeter of the base times the height,  where  is equal to the area of the base and top,  is the perimeter of the base, and  is the height.
The formula for the volume of a rectangular prism, given in cubic units, is equal to the area of the base times the height,  where  is the area of the base and  is the height.
The surface area of a right cylinder is given as
The volume of a right cylinder is given as


# Geometry

## Right Triangle Trigonometry

### Learning Objectives

After completing this section, you should be able to:
1. Apply the Pythagorean Theorem to find the missing sides of a right triangle.
2. Apply the  and  right triangle relationships to find the missing sides of a triangle.
3. Apply trigonometric ratios to find missing parts of a right triangle.
4. Solve application problems involving trigonometric ratios.
This is another excerpt from Raphael’s The School of Athens. The man writing in the book represents Pythagoras, the namesake of one of the most widely used formulas in geometry, engineering, architecture, and many other fields, the Pythagorean Theorem. However, there is evidence that the theorem was known as early as 1900–1100 BC by the Babylonians. The Pythagorean Theorem is a formula used for finding the lengths of the sides of right triangles.
Born in Greece, Pythagoras lived from 569–500 BC. He initiated a cult-like group called the Pythagoreans, which was a secret society composed of mathematicians, philosophers, and musicians. Pythagoras believed that everything in the world could be explained through numbers. Besides the Pythagorean Theorem, Pythagoras and his followers are credited with the discovery of irrational numbers, the musical scale, the relationship between music and mathematics, and many other concepts that left an immeasurable influence on future mathematicians and scientists.
The focus of this section is on right triangles. We will look at how the Pythagorean Theorem is used to find the unknown sides of a right triangle, and we will also study the special triangles, those with set ratios between the lengths of sides. By ratios we mean the relationship of one side to another side. When you think about ratios, you should think about fractions. A fraction is a ratio, the ratio of the numerator to the denominator. Finally, we will preview trigonometry. We will learn about the basic trigonometric functions, sine, cosine and tangent, and how they are used to find not only unknown sides but unknown angles, as well, with little information.

### Pythagorean Theorem

The Pythagorean Theorem is used to find unknown sides of right triangles. The theorem states that the sum of the squares of the two legs of a right triangle equals the square of the hypotenuse (the longest side of the right triangle).
For example, given that side  and side  we can find the measure of side  using the Pythagorean Theorem. Thus,

### Distance

The applications of the Pythagorean Theorem are countless, but one especially useful application is that of distance. In fact, the distance formula stems directly from the theorem. It works like this:
In , the problem is to find the distance between the points  and  We call the length from point  to point  side , and the length from point  to point  side . To find side , we use the distance formula and we will explain it relative to the Pythagorean Theorem. The distance formula is  such that  is a substitute for  in the Pythagorean Theorem and is equal to  and  is a substitute for  in the Pythagorean Theorem and is equal to  When we plug in these numbers to the distance formula, we have

Thus, , the hypotenuse, in the Pythagorean Theorem.

###  Triangles

In geometry, as in all fields of mathematics, there are always special rules for special circumstances. An example is the perfect square rule in algebra. When expanding an expression like  we do not have to expand it the long way:

If we know the perfect square formula, given as

    we can skip the middle step and just start writing down the answer. This may seem trivial with problems like  However, what if you have a problem like  That is a different story. Nevertheless, we use the same perfect square formula. The same idea applies in geometry. There are special formulas and procedures to apply in certain types of problems. What is needed is to remember the formula and remember the kind of problems that fit. Sometimes we believe that because a formula is labeled special, we will rarely have use for it. That assumption is incorrect. So, let us identify the  triangle and find out why it is special. See .
We see that the shortest side is opposite the smallest angle, and the longest side, the hypotenuse, will always be opposite the right angle. There is a set ratio of one side to another side for the  triangle given as  or  Thus, you only need to know the length of one side to find the other two sides in a  triangle.

###  Triangles

The  triangle is another special triangle such that with the measure of one side we can find the measures of all the sides. The two angles adjacent to the  angle are equal, and each measures  If two angles are equal, so are their opposite sides. The ratio among sides is  or  as shown in .

### Trigonometry Functions

Trigonometry developed around 200 BC from a need to determine distances and to calculate the measures of angles in the fields of astronomy and surveying. Trigonometry is about the relationships (or ratios) of angle measurements to side lengths in primarily right triangles. However, trigonometry is useful in calculating missing side lengths and angles in other triangles and many applications.
Trigonometry is based on three functions. We title these functions using the following abbreviations:

Letting  which is the hypotenuse of a right triangle, we have . The functions are given in terms of , , and , and in terms of sides relative to the angle, like opposite, adjacent, and the hypotenuse.
We will be applying the sine function, cosine function, and tangent function to find side lengths and angle measurements for triangles we cannot solve using any of the techniques we have studied to this point. In , we have an illustration mainly to identify  and the sides labeled  and .
An angle  sweeps out in a counterclockwise direction from the positive -axis and stops when the angle reaches the desired measurement. That ray extending from the origin that marks  is called the terminal side because that is where the angle terminates. Regardless of the information given in the triangle, we can find all missing sides and angles using the trigonometric functions. For example, in , we will solve for the missing sides.
Let’s use the trigonometric functions to find the sides  and . As long as your calculator mode is set to degrees, you do not have to enter the degree symbol. First, let’s solve for .
We have  and  Then,

Next, let’s find . This is the cosine function. We have  Then,

Now we have all sides,  You can also check the sides using the  ratio of   is a list of common angles, which you should find helpful.
To find angle measurements when we have two side measurements, we use the inverse trigonometric functions symbolized as   or  The –1 looks like an exponent, but it means inverse. For example, in the previous example, we had  and  To find what angle has these values, enter the values for the inverse cosine function  in your calculator:

You can also use the inverse sine function and enter the values of  in your calculator given  and  We have

Finally, we can also use the inverse tangent function. Recall  We have

### Angle of Elevation and Angle of Depression

Other problems that involve trigonometric functions include calculating the angle of elevation and the angle of depression. These are very common applications in everyday life. The angle of elevation is the angle formed by a horizontal line and the line of sight from an observer to some object at a higher level. The angle of depression is the angle formed by a horizontal line and the line of sight from an observer to an object at a lower level.

### Check Your Understanding

### Key Terms

1. right triangle
2. sine
3. cosine
4. tangent

### Key Concepts

1. The Pythagorean Theorem is applied to right triangles and is used to find the measure of the legs and the hypotenuse according the formula  where c is the hypotenuse.
2. To find the measure of the sides of a special angle, such as a  triangle, use the ratio  where each of the three sides is associated with the opposite angle and 2 is associated with the hypotenuse, opposite the  angle.
3. To find the measure of the sides of the second special triangle, the  triangle, use the ratio  where each of the three sides is associated with the opposite angle and  is associated with the hypotenuse, opposite the  angle.
4. The primary trigonometric functions are   and
5. Trigonometric functions can be used to find either the length of a side or the measure of an angle in a right triangle, and in applications such as the angle of elevation or the angle of depression formed using right triangles.

### Formula

The Pythagorean Theorem states

    where  and  are two sides (legs) of a right triangle and  is the hypotenuse.

### Projects

1. One of the reasons so many formulas in geometry were discovered was because of the importance in finding measurements of lengths, areas, perimeter, and angles. Find at least five examples of how geometry can be used in practical applications today.
2. Who were the Pythagoreans? Why did this society exist? Explore what they did and discuss some of their beliefs.

### Chapter Review

### Points, Lines, and Planes

### Angles

### Triangles

### Polygons, Perimeter, and Circumference

### Tessellations

### Area

### Volume and Surface Area

### Right Triangle Trigonometry

### Chapter Test


# Voting and Apportionment

## Introduction

Suppose a friend asked you, “When did you last vote?” What would your answer be? Maybe you would tell your friend that the last time you voted was during the last presidential election, or perhaps you would tell your friend that you prefer not to vote. When thinking about voting, presidential campaigns or advertisements for reelections may come to mind, but you can cast your vote in many ways. Have you liked a post, followed a creator, friended a stranger, or clicked a heart online today? In the digital age, it's possible to vote several times a day. Voting systems are not only the machines that drive every democracy on Earth, but they are also the engines driving social media and many other aspects of life. A deeper understanding of these voting systems will enhance your ability to successfully engage with the world in which we live.
In this chapter, you will become one of the founders of the new democratic country of Imaginaria. You have a great responsibility to the people of this fledgling democracy. You have been tasked with writing the portion of the constitution that lays out voting procedures. In preparation for this important task, you will explore the various types of voting systems, from school board elections to Twitter wars. You will see how these types are alike, how they differ, and how they might be applied in Imaginaria. Most importantly, you will learn about the mathematically inherent advantages and disadvantages of various voting systems so that you can make informed choices to better the lives of the Imaginarians.


# Voting and Apportionment

## Voting Methods

### Learning Objectives

After completing this section, you should be able to:
1. Apply plurality voting to determine a winner.
2. Apply runoff voting to determine a winner.
3. Apply ranked-choice voting to determine a winner.
4. Apply Borda count voting to determine a winner.
5. Apply pairwise comparison and Condorcet voting to determine a winner.
6. Apply approval voting to determine a winner.
7. Compare and contrast voting methods to identify flaws.
Today is the day that you begin your quest to collaborate on the constitution of Imaginaria! Let’s begin by thinking about the selection of a leader who can serve as president. It seems straightforward; if the majority of citizens prefer a particular candidate, that candidate should win. But not all votes are decided by a simple majority. Why not? What are the options?

### Majority versus Plurality Voting

When an election involves only two options, a simple majority is a reasonable way to determine a winner. A majority is a number equaling more than half, or greater than 50 percent of the total.
Let’s take a look at the outcomes of U.S. presidential elections to understand more.  displays the results of the 2000 U.S. presidential election. Like most presidential elections, this election involved more than two options. If that is the case, is it possible that no single candidate will receive more than half of the votes cast?
Unlike in the 2000 U.S. presidential election, a candidate won the majority of votes in the 2020 election (see ). It is a common occurrence for no single candidate to receive a majority of the votes in an election with more than two candidates. When this occurs, the candidate with the largest portion of the votes is said to have a plurality.
Your plans for Imaginarian elections will likely include primary elections, or preliminary elections to select candidates for a principal or general election.  displays the results of the 2018 U.S. Senate Republican primary for Maryland.
Consider how election by plurality, not majority, is the most common method of selecting candidates for public office.

### Runoff Voting

Has your family ever debated what to have for dinner? Suppose your family is deciding on a restaurant and exactly half of you want to have pizza but the other half want hamburgers. How do you decide when the result is a tie? You need a tiebreaker!
Will the new democracy of Imaginaria need tiebreakers? When no candidate satisfies the requirements to win the election, a runoff election, or second election, is held to determine a winner.
How would runoff voting work in Imaginaria? There are many types of runoff voting systems, which are voting systems that utilize a runoff election when the first round does not result in a winner. The method for implementing a runoff election can vary widely, particularly in the criteria used to determine whether a candidate will be on the ballot in the second election. For example, a two-round system is a runoff voting system in which only the top candidates advance to the runoff election. In some two-round systems, only the top two candidates are on the second ballot, or it may be any candidate who secures a certain percentage of the vote will advance. The Hare Method is another runoff voting system in which only the candidate(s) with the very least votes are eliminated. This can potentially result in several rounds of runoff elections.

### Steps to Determine Winner by Plurality or Majority Election with Runoff

To determine the winner by plurality or when a majority election with runoff occurs, we take these three steps:
Step 1: If a majority is required to win the election, determine the number of votes needed to achieve a majority. This is the least whole number greater than 50 percent of the total votes. If a majority is not required, move to Step 2.
Step 2: Count the number of votes for each candidate in the current round of voting. If a single candidate has enough votes to win a plurality, or a majority as appropriate, then you are done! Otherwise, eliminate a predetermined number of candidates based on the rules of the election. Elimination conditions may vary. For example, the rules may state that the candidate(s) with the fewest votes will be eliminated (as in the Hare method), or that only the candidates meeting a certain threshold will move on (as in a two-round system). Once the appropriate candidates are eliminated, move on to Step 3.
Step 3: Hold a runoff election. If the runoff is simulated using a list of voter’s preferences, renumber the preferences to reflect the remaining number of options in such a way that the original order of preference is retained. Then repeat Step 2.
Note: The second and third steps may be repeated as many times as necessary for voting procedures that allow multiple runoffs.

### Ranked-Choice Voting

In  and Your Turn 11.4, you were given a list that ordered each voter’s preferences. This ordering is called a preference ranking. A ballot in which a voter is required to give an ordering of their preferences is a ranked ballot, and any voting system in which a voter uses a ranked ballot is referred to as ranked voting.
The vote for the Academy Awards uses a ranked ballot. The table below provides an example of a ranked ballot for the 2020 Academy Award nominees for Best Director.
As you decide on the voting methods that will be used in your new democracy, budget must be a consideration. You might consider a particular type of ranked voting called ranked-choice voting (RCV), which simulates a series of runoff elections without the usual time and expense involved when voters must repeatedly return to the polls, like we did in .
The method of ranked-choice voting (RCV), also called instant runoff voting (IRV), is a version of the Hare Method, using preference ranking so that, if no single candidate receives a majority, the least popular selections can be eliminated and the results can be recounted, without the need for more elections.
As we explore examples of ranked voting, we will summarize the voters’ preference rankings using a table in which the top row shows the number of ballots that ranked the options in the same order. Let’s practice interpreting the information in this type of table.
Now that we’ve covered how to read a summary of preference rankings, let’s practice using the ranked-choice method to determine the winner of an election. Recall that ranked-choice voting is still the Hare Method where the candidate with the very least number of votes is eliminated each round until a majority is attained. The difference here is that the voters have completed a ranked ballot, so they don't have to visit the polls multiple times. Here are the steps for ranked-choice voting.

### Steps to Determine Winner by Ranked-Choice Voting

To determine the winner when ranked-choice voting occurs, we take these three steps:
Step 1: Determine the number of votes needed to achieve a majority. This is the least whole number greater than 50 percent of the total votes.
Step 2: Count the number of first place votes for each candidate. If a candidate has a majority, that candidate wins the election and we are done! Otherwise, eliminate the candidate(s) with the fewest votes and complete Step 3.
Step 3: Reallocate the votes to the remaining candidates, and repeat Step 2.

### Borda Count Voting

Ranked-choice voting is one type of ranked voting that simulates multiple runoffs based on ranked ballots. Another type of ranked voting is the Borda count method, which uses ranked ballots that award candidates points corresponding to the number of candidates ranked lower on each ballot.
To understand how this works, let’s review the favorite colors of our kindergarten class from the table below. Let’s focus on the votes represented by the first column of the preference summary.
Each student had six options. This first column tells us that four students ranked blue as their first choice, red as their second choice, pink as their third choice, purple as their fourth choice, yellow as their fifth choice, and green as their sixth choice. Blue was ranked higher than  other colors. For each of the four students who completed their ballot in this way, blue would receive five points. Since there were four ballots with this ordering, blue would receive
    points from the first column. To determine the total points for each candidate, we have to find the sum of the points they received in each column.
To determine the winner of a contest using the Borda count method, we must compare total number of points earned by each candidate. The candidate with the most points is the winner. Each row of the preference summary corresponds to a single candidate. To find the number of points received by a particular candidate in the preference summary, or their Borda score, we will need to focus on the row in which that candidate appears.
Before we practice determining the winner of a Borda count election, let’s examine how to find the Borda score for a single candidate.
Now let’s determine the winner of an election by comparing the Borda scores for each of the candidates.
The Borda count method may seem too complicated to even consider using for Imaginaria, but each voting method has its own pros and cons. The Borda count method, for example, favors compromise candidates over divisive candidates. A compromise candidate is not the first choice of most of the voters, but is more acceptable to the population as a whole than the other candidates. A divisive candidate is simultaneously the first choice of a large portion of the voters and the last choice of another large portion of the voters.
In , Candidate A was ranked first by 225 voters, but was ranked last by 185 voters. No voters ranked Candidate A as second or third. It appears that, although Candidate A had the majority of first place votes, there was a significant minority who strongly disliked them. Candidate A was a divisive candidate. Candidate B, on the other hand, was the second choice of every voter, making Candidate B a good compromise. The Borda count method chose Candidate B, a compromise candidate, that was more acceptable to the population as a whole. This scenario is cited by both opponents and proponents of the Borda count method.

### Pairwise Comparison and Condorcet Voting

We have discussed two kinds of ranked voting methods so far: ranked-choice and Borda count. A third type of ranked voting is the pairwise comparison method, in which the candidates receive a point for each candidate they would beat in a one-on-one election and half a point for each candidate they would tie. If one candidate earns more points than the others, then that candidate wins. This method is one of several Condorcet voting methods, which are methods in which candidates are ranked and then compared pairwise to each other, a candidate having to beat all others in order to win. These methods vary in the way candidates are scored, and there is not always a clear winner. A candidate who wins each possible pairing is known as a Condorcet candidate. These terms are named after the Marquis de Condorcet, a French philosopher and mathematician who preferred the pairwise comparison method to the Hare method and made public arguments in its favor.
If you include a Condorcet voting method in the constitution of Imaginaria, the election supervisors may want to use a pairwise comparison matrix like the one in . It’s a tool used to list the number of wins associated with each pairing of two candidates. Each candidate will receive a point for each win and a half a point for each tie. Each pairing is listed twice, once for the number of wins of a candidate over a particular challenger and once for the number of wins of the challenger over that candidate.

### Steps to Determine a Winner by Pairwise Comparison Method Using a Matrix

To determine the winner when the pairwise comparison method is used, we take these three steps:
Step 1: On the matrix, indicate a losing matchup by crossing out a box, , and tie match ups by drawing a slash through the box, .
Step 2: Award each candidate 1 point for a win, half a point for a tie, and 0 points for a loss.
Step 3: Identify the winner, which is the candidate with the most points.
Before you decide on the pairwise comparison method for Imaginaria, review what’s involved in constructing a pairwise comparison matrix from a summary of ranked ballots. Then we can use the matrix to determine the winner of the election. Does the winner using the Borda method still win?

### Three Key Questions

Before you decide if you want to use the pairwise comparison method for Imaginarian elections, let’s consider three questions that might affect your decision.
1. Is there always a winner?
2. If there is a winner, is the winner always a Condorcet candidate?
3. If there is a Condorcet candidate, does that candidate always win?
Let’s think about why these questions might be important to you if you chose the pairwise comparison method. First, if no candidate meets the criteria to win an election, you will need a backup plan such as a runoff election. Second, if the winner is not a Condorcet candidate, then there is at least one candidate who beat the winner in a pairwise matchup and the supporters of that candidate might question the validity of the election. Finally, if there is a Condorcet candidate who beat every other candidate in a pairwise matchup, it is reasonable to conclude that it would be unfair for anyone else to win. The rest of the examples in this section should illustrate these key concepts.
 illustrates the answer to the first key question. The pairwise comparison method does not always result in a winner. For example, much like the game of Rock, Paper, Scissors, it is possible for a cyclic pattern to emerge in which each candidate beats the next until the last candidate who beats the first.
Now, you have the answer to the second key question. The pairwise comparison matrix in YOUR TURN 11.10 is an example of a scenario where a winner is not a Condorcet candidate.
The answer to the third question is not as clear. If there is a Condorcet candidate, does that candidate win? So far, we have not come across a contradictory example where the Condorcet candidate didn't win, but we cannot know with certainty that it is not possible by looking at examples. Instead, we will need to use some reasoning. Let’s review some particular cases of elections with a certain number of candidates, and then we will try to generalize the scenario to an election with  candidates.
Let’s consider a general case where there are  candidates. One of the candidates is a Condorcet candidate. Since the Condorcet candidate wins all matchups, the Condorcet candidate wins  points. Since each of the other candidates lost to the Condorcet candidate, the most a single candidate could win is . Since the Condorcet candidate won  points and each other candidate won  points or fewer, the Condorcet candidate is the winner. You have your answer to the third key question! If there is a Condorcet candidate, that candidate is always the winner.

### Approval Voting

The last type of voting system you will consider for your budding democracy is an approval voting system. In this system, each voter may approve any number of candidates without rank or preference for one over another (among the approved candidates), and the candidate approved by the most voters wins.
    This voting system has aspects in common with plurality voting and Condorcet voting methods, but it has characteristics that distinguish it from both. An approval voting ballot lists the candidates and provides the option to approve or not approve each candidate.
The term “approval voting” was not used until the 1970s Brams, Steven J.; Fishburn, Peter C. (2007), Approval Voting, Springer-Verlag, p. xv, ISBN 978-0-387-49895-9, although its use has been documented as early as the 13th century (Brams, Steven J. (April 1, 2006). The Normative Turn in Public Choice (PDF) (Speech). Presidential Address to Public Choice Society. New Orleans, Louisiana.) Approval voting has the appeal of being simpler than ranked voting methods. It also allows an individual voter to support more than one candidate equally. This has appeal for those who do not want a split vote among a few mainstream candidates to lead to the election of a fringe candidate. It also has appeal for those who want an underdog to have a chance of success because voters will not worry about wasting their vote on a candidate who is not believed likely to win.

### Compare and Contrast Voting Methods to Identify Flaws

Wow! We have covered a lot of options for the voting methods. Now, you need to decide which one is best for Imaginaria. Imaginarians might consider characteristics of certain voting systems desirable and others undesirable. In some cases, voters may consider these undesirable traits to be flaws in a voting system that are significant enough to motivate them to reject that system. If you are feeling a bit overwhelmed by this decision, maybe it would help to read about the experiences of others who have faced similar questions.
Consider the 2000 U.S. presidential election in which Green Party candidate Ralph Nader and Reform Party candidate Pat Buchanan were on the ballet running against the mainstream candidates, Democrat Al Gore and Republican George W. Bush. The voting results for Florida are given in .
In more than one state, Buchanan was able to split the Republican vote enough to allow Gore to win that state. Nader split the Democrat vote in Florida and New Hampshire by enough votes to prevent Gore from winning those states. Had Gore won either state, he would have had enough electoral votes to win the election. Instead, Bush won. This is an example of a flaw in the plurality system of voting: the spoiler.
A spoiler is a less popular candidate who takes votes from a more popular candidate with similar positions, swinging the race to another candidate with vastly different views that they would not support. This encourages voters not to vote for the candidate that they perceive to be the best, but instead for the candidate they can live with who they perceive to have a better chance of winning. Some voters may prefer a method such as approval voting, which does not have this trait in common with plurality voting.
The results in  and Your Turn 11.14 highlight one of the characteristics of approval voting. Ralph Nader moved up from a distant third place finish to a close second place finish when Al Gore’s supporters approved him on their ballots. In this way, fringe candidates have a better chance of winning, which some voters consider a flaw but others consider a benefit.
Another aspect of approval voting systems that is a concern to many voters is that candidates in approval elections might encourage their loyal supporters to approve them and only them to avoid giving support to any other candidate. If this occurred, the election in effect becomes a traditional plurality election. This is a flaw that cannot occur in an instant runoff system since all candidates are ranked.
The election in  involves a scenario in which there are two extreme candidates, Planet A and Planet B, and a moderate candidate, Planet C. The supporters of the extreme candidates prefer the moderate candidate to the other extremist ones. This makes Planet C a compromise candidate. In this case, both the plurality method and ranked-choice voting resulted in the election of one of the extreme candidates, but the Borda count method elected the compromise candidate in this scenario. Depending on a person’s perspective, this may be perceived as a flaw in either ranked-choice and plurality systems, or the Borda count method.
In Fairness in Voting Methods, we will analyze the fairness of each voting system in greater detail using objective measures of fairness.

### Check Your Understanding

### Key Terms

1. majority
2. plurality
3. runoff election
4. runoff voting system
5. two-round system
6. Hare Method
7. preference ranking
8. ranked ballot
9. ranked-choice voting (RCV)
10. instant runoff voting (IRV)
11. Borda count method
12. Borda score
13. Compromise candidate
14. divisive candidate
15. pairwise comparison method
16. Condorcet voting methods
17. Condorcet candidate
18. approval voting system
19. approval voting ballot
20. spoiler

### Key Concepts

1. In plurality voting, the candidate with the most votes wins.
2. When a voting method does not result in a winner, runoff voting can be used to do so.
3. Ranked-choice voting, also known as instant runoff voting, is one type of ranked voting system.
4. The Borda count method is a type of ranked voting system in which each candidate is given a Borda score based on the number of candidates ranked lower than them on each ballot.
5. When pairwise comparison is used, the winner will be the Condorcet candidate if one exists.
6. Approval voting allows voters to give equally weighted votes to multiple candidates.
7. When a voter finds a characteristic of a particular voting method unappealing, they may consider that characteristic a flaw in the voting method and look for an alternative method that does not have that characteristic.

### Videos

1. How Does Ranked-Choice Voting Work?
2. Determine Winner of Election by Ranked-Choice Method (aka Instant Runoff)
3. Determine Winner of Election by Borda Count Method
4. Determine Winner of Election by Pairwise Comparison Method


# Voting and Apportionment

## Fairness in Voting Methods

### Learning Objectives

After completing this section, you should be able to:
1. Compare and contrast fairness of voting using majority criterion.
2. Compare and contrast fairness of voting using head-to-head criterion.
3. Compare and contrast fairness of voting using monotonicity criterion.
4. Compare and contrast fairness of voting using irrelevant alternatives criterion.
5. Apply Arrow’s Impossibility Theorem when evaluating voting fairness.
Now that we’ve covered a variety of voting methods and discussed their differences and similarities, you might be leaning toward one method over another. You will need to convince the other founders of Imaginaria that your preference will be the best for the country. Before your collaborators approve the inclusion of a voting method in the constitution, they will want to know that the voting method is a fair method. In this section, we will formally define the characteristics of a fair system. We will analyze each voting previously discussed to determine which characteristics of fairness they have, and which they do not. In order to guarantee one ideal, we must often sacrifice others.

### The Majority Criterion

One of the most fundamental concepts in voting is the idea that most voters should be in favor of a candidate for a candidate to win, and that a candidate should not win without majority support. This concept is known as the majority criterion.
With respect to the four main ranked voting methods we have discussed—plurality, ranked-choice, pairwise comparison, and the Borda count method—we will explore two important questions:
1. Which of these voting systems satisfy the majority criterion and which do not?
2. Is it always “fair” for a voting system to satisfy the majority criterion?
Keep in mind that this criterion only applies when one of the candidates has a majority. So, the examples we will analyze will be based on scenarios in which a single candidate has more than 50 percent of the vote.
From , it appears that the plurality and ranked-choice voting methods satisfy the majority criterion. In general, the majority candidate always wins in a plurality election because the candidate that has more than half of the votes has more votes than any other candidate. The same is true for ranked-choice voting; and there will never be a need for a second round when there is a majority candidate. Let’s examine how some of the other voting methods stand up to the majority criterion.
 demonstrates a concept that we also saw in Borda Count Voting—the Borda method frequently favors the compromise candidate over the divisive candidate. This can happen even when the divisive candidate has a majority, as it did in this example. Although a majority of the voters were in favor of McDonald’s, a significant minority was strongly opposed to McDonald’s, ranking it last. Since the Borda score includes all rankings, this strong opposition has an impact on the outcome of the election.
Pairwise comparison will always satisfy the majority criterion because the candidate with the majority of first-place votes wins each pairwise matchup. While it is possible for the majority candidate to win by the Borda count method, it is not guaranteed. So, the Borda method fails the majority criterion. A summary of each voting method as it relates to the majority criterion is found in the following table.
If you prefer the Borda method, you might argue that its failure to satisfy the majority criterion is actually one of its strengths. As we saw in  and , the majority have the power to vote for their own benefit at the expense of the minority. While four students were very enthusiastic about McDonald’s, three students were strongly opposed to McDonald’s. It is reasonable to say that the better option would be Burger King, the compromise candidate, which everyone ranked highly and no one strongly opposed. The Borda method is designed to favor a candidate that is acceptable to the population as a whole. In this way, the Borda method avoids a downfall of strict majority rule known as the tyranny of the majority, which occurs when a minority of a population is treated unfairly because their situation is different from the situation of the majority.
The people of Imaginaria should know that the power of the majority to vote their will has serious implications for other groups. For example, according to the UCLA School of Law Williams Institute, the LGBTQ+ community in the United States makes up approximately 4.5 percent of the population. When elections occur that include issues that affect the LGBTQ+ community, members of the LGBTQ+ community depend on the 95.5 percent of the population who do not identify as LGBTQ+ to consider their perspectives when voting on issues such as same-sex marriage, the use of public restrooms by transgender people, and adoption by same-sex couples.

### Head-to-Head Criterion

Another fairness criterion you must consider as you select a voting method for Imaginaria is the Condorcet criterion, also known as the head-to-head criterion. An election method satisfies the Condorcet criterion provided that the Condorcet candidate wins the election whenever a Condorcet candidate exists. A Condorcet method is any voting method that satisfies the Condorcet criterion.
Recall from Three Key Questions that not every election has a Condorcet candidate; the Condorcet criterion will not apply to every election. Also recall that a Condorcet candidate cannot lose an election by pairwise comparison. So, the pairwise comparison voting method is said to satisfy the Condorcet criterion.
As we have seen, the plurality method, ranked-choice voting, and the Borda count method each fail the Condorcet criterion in some circumstances. Of the four main ranked voting methods we have discussed, only the pairwise comparison method satisfies the Condorcet criterion every time. A summary of each voting method as it relates to the Condorcet criterion is found in the following table.

### Monotonicity Criterion

The citizens of Imaginaria might be surprised to learn that it is possible for a voter to cause a candidate to lose by ranking that candidate higher on their ballot. Is that fair? Most voters would say, “Absolutely not!!” This is an example of a violation of the fairness criterion called the monotonicity criterion, which is satisfied when no candidate is harmed by up-ranking nor helped by down-ranking, provided all other votes remain the same.
Consider a scenario in which voters are permitted a first round that is not binding, and then they may change their vote before the second round. Such a first round can be called a “straw poll.” Now, let’s suppose that a particular candidate won the straw poll. After that, several voters are convinced to increase their support, or up-rank, that winning candidate and no voters decrease that support. It is reasonable to expect that the winner of the first round will also win the second. Similarly, if some of the voters decide to decrease their support, or down-rank, a losing candidate, it is reasonable to expect that candidate will still lose in the second round.
You might be wondering why it’s called the monotonicity criterion. In mathematics, the term monotonicity refers to the quality of always increasing or always decreasing. For example, a person’s age is monotonic because it always increases, whereas a person’s weight is not monotonic because it can increase or decrease. If the only changes to the votes for a particular candidate after a straw poll are in one direction, this change is considered monotonic.
If you are going to make an informed decision about which voting method to use in Imaginaria, you need to know which of the four main ranked voting methods we have discussed—plurality, ranked-choice, pairwise comparison, and the Borda count method—satisfy the monotonicity criterion.
The last few examples illustrate that the plurality method, pairwise comparison voting, and the Borda count method each satisfy the monotonicity criterion. Of the four main ranked voting methods we have discussed, only the ranked-choice method violates the monotonicity criterion. A summary of each voting method as it relates to the Condorcet criterion is found in the table below.

### Irrelevant Alternatives Criterion

We have covered a lot about voting fairness, but there is one more fairness criterion that you and the other Imaginarians should know. Consider this well-known anecdote that is sometimes attributed to the American philosopher Sidney Morgenbesser:
A man is told by his waiter that the dessert options this evening are blueberry pie or apple pie. The man orders the apple pie. The waiter returns and tells him that there is also a third option, cherry pie. The man says, “In that case, I would like the blueberry pie.” (Gaming the Vote: Why Elections Aren’t Fair (and What We Can Do About It), William Pound stone, p. 50, ISBN 0-8090-4893-0)
This story illustrates the concept of the Irrelevant Alternatives Criterion, also known as the Independence of Irrelevant Alternatives Criterion (IIA), which means that the introduction or removal of a third candidate should not change or reverse the rankings of the original two candidates relative to one another. In particular, if a losing candidate is removed from the race or if a new candidate is added, the winner of the race should not change.
We have seen that all four of the main voting systems we are working with fail the Irrelevant Alternatives Criterion (IIA). A summary of each voting method as it relates to the IIA criterion is found in the table below.
You might be wondering if there is a voting system you could recommend for Imaginaria that satisfies all the fairness criteria. If there is one, it remains to be discovered and it is not a voting system that is based solely on preference rankings. In 1972, Harvard Professor of Economics Kenneth J. Arrow received the Nobel Prize in Economics for proving Arrow’s Impossibility Theorem, which states that any voting system, either existing or yet to be created, in which the only information available is the preference rankings of the candidates, will fail to satisfy at least one of the following fairness criteria: the majority criterion, the Condorcet criterion, the monotonicity criterion, and the independence of irrelevant alternatives criterion. This theorem only applies to a specific category of voting systems—those for which the preference ranking is the only information collected. There are other types of voting systems to which the Impossibility Theorem does not apply. For example, there is a class of voting systems called Cardinal voting systems that allow for rating the candidates in some way.
“Rating” is different from “ranking” because a voter can give different candidates the same rating. Consider the five-star rating systems used by various industries, or the thumbs up/thumbs down rating system used on YouTube. Could there be a Cardinal voting system that does not violate any of the fairness criteria we have discussed? It’s possible, but more research must be done in order to prove it.

### Check Your Understanding

### Key Terms

1. majority criterion
2. tyranny of the majority
3. Condorcet criterion
4. Condorcet method
5. monotonicity criterion
6. up-rank
7. down-rank
8. independence of irrelevant alternatives criterion (IIA)
9. Arrow’s Impossibility Theorem
10. cardinal voting system

### Key Concepts

1. There are several common measures of voting fairness, including the majority criterion, the head-to head criterion, the monotonicity criterion, and the irrelevant alternatives criterion.
2. According to Arrow’s Impossibility Theorem, each voting method in which the only information is the order of preference of the voters will violate one of the fairness criteria.

### Videos

1. Separation of Powers and Checks and Balances


# Voting and Apportionment

## Standard Divisors, Standard Quotas, and the Apportionment Problem

### Learning Objectives

After completing this section, you should be able to:
1. Analyze the apportionment problem and applications to representation.
2. Evaluate applications of standard divisors.
3. Evaluate applications of standard quotas.

### The Apportionment Problem

In the new democracy of Imaginaria, there are four states: Fictionville, Pretendstead, Illusionham, and Mythbury. Each state will have representatives in the Imaginarian Legislature. You might now have an agreement on which voting method your citizens will use to elect representatives. However, before that process can even begin, you must decided on how many representatives each state will receive. This decision will present its own challenges.
When sharing your birthday cake, it’s only fair that everyone gets the same portion size, right? You were portioning the cake by dividing it up equally and giving everyone a slice. A great thing about cake is that you can slice it any way you want, but how do you apportion, or divide and distribute, items that can't be sliced? Suppose that you have a box of 16 Ring Pops™, gem-shaped lollipops on a plastic ring. You are going to share the box with four other kids. Dividing the 16 Ring Pops™ among the group of five leads to a problem; after each person in the group gets three Ring Pops™, there is still one left! Who gets the last one? The apportionment problem is how to fairly divide, or apportion, available resources that must be distributed to the recipients in whole, not fractional, parts.
The apportionment problem applies to many aspects of life, including the representatives in the Imaginarian legislature. The table below provides a short list of examples of resources that must be apportioned in whole parts, and the recipients of those resources.
Fair division of a resource is not necessarily equal division of the resource like when distributing cake slices. When distributing airport terminals amongst airlines, there are many factors to consider such as the size of the airline, the number and types of aircraft they have, and the demand for the service. In most cases, fairness is defined as being proportional; two quantities are proportional if they have the same relative size. In the case of the Covid-19 vaccine, the expectation would be that countries with larger populations get more doses of the vaccine. In the Imaginarian legislature, the expectation may be that the states with larger populations will receive the larger number of representatives. This concept is referred to as a part-to-part ratio.
Suppose that a supermarket has a special on pies, two for $5. The first customer purchases four pies for $10, and the second customer purchases eight pies for $20. The dollar to pie ratio for the first customer is  and the dollar to pie ratio for the second customer is . So, the dollar to pie ratio is constant. Although the customers do not spend the same amount of money, the amount each spent was proportional to the number of pies purchased.
Now suppose that the supermarket changed the special to $5 for the first pie, and $2 for each additional pie. In that case, four pies would cost , while 8 pies would cost . The dollar to pie ratios would be  and , respectively. This special does not result in a constant part to part ratio. The dollars spent are not proportional to the number of pies purchased.
There are some useful relationships between quantities that are proportional to each other. When there is a constant ratio between two quantities, the one quantity can be found by multiplying the other by that ratio. Remember the supermarket special on pies, 2 pies for $5? The ratio of dollars to pies is  and the ratio of pies to dollars is. These two values are reciprocals of each other,  and . This means that multiplying by one has the same effect as dividing by the other. This also means that knowing either constant ratio allows us to calculate the price given the number of pies. To find the cost of 20 pies, multiply by the ratio of dollars to pies or divide by the ratio of pies to dollars.
1.
2.
These patterns are true in general.
The apportionment application that will be important to the founders of Imaginaria occurs in representative democracies in which elected persons represent a group. The United Kingdom, France, and India each have a parliament, and the United States has a Congress, just as Imaginaria will have a legislature! The citizens of a country must decide what portion of the representatives each group, such as a state or province or even a political party, will have. A larger portion of representatives means greater influence over policy.
You might be wondering why the ratio doesn't appear to be quite the same depending on the rounding of the values. We will see that the key to this variation lies in the fractions. Just like the five children sharing 16 Ring Pops™, there are going to be leftovers and there are many methods for deciding what to do with those leftovers.

### The Standard Divisor

There are two houses of congress in the United States: the Senate and the House of Representatives. Each state has two senators, but the number of representatives depends on the population of the state. The number of representative seats in the U.S. House of Representatives is currently set by law to be 435. In order to distribute the seats fairly to each state, the ratio of the population of the U.S. to the number of representative seats must be calculated. The ratio of the total population to the house size is called the standard divisor, and it is the number of members of the total population represented by one seat.
Although apportionment applies to many other scenarios, such as the pencil distribution during the SAT, the terminology of apportionment is based on the House of Representatives scenario. Thus, several government-related terms take on a more general meaning. The states are the recipients of the apportioned resource, the seats are the units of the resource being apportioned, the house size is the total number of seats to be apportioned, the state population is the measurement of the state's size, and the total population is the sum of the state populations.
Whether the standard divisor is less than, equal to, or greater than 1 depends on the ratio of the population to the number of seats.
1. The standard divisor will be equal to 1 if the total population is equal to the number of seats. This would mean that each member of the population is allocated their own personal seat.
2. The standard divisor will be a number between 0 and 1 when the total population is less than the number of seats. This means that each member of the population is allocated more than one seat.
3. The standard divisor will be a number greater than 1when the total population is greater than the number of seats. This means that a certain number of members of the population will share 1 seat.
If the population is five children and the house consists of five pieces of candy, the standard divisor is  meaning each child gets one candy. If the population is five children and ten pieces of candy, the standard divisor is  meaning that each child gets more than one candy. If the population is five children and four pieces of candy, the standard divisor is  meaning that each child gets less than one candy.
If the seats in the Imaginarian legislature are distributed to the states based on population, then the house size will be less than the population and we should expect the standard divisor to be a number greater than 1.

###  The Standard Quota

Once the standard divisor for the Imaginarian legislature is calculated, the next task is to determine the number of seats that each state should receive, which is referred to as the state’s standard quota. Unless all the states have the same population, each state will receive a different number of seats because the quantities will be proportionate to the state populations. To determine those amounts, we will use an idea we learned earlier. Recall that, when the number of units of item  is proportionate to the number of units of item , we have:
In this case, we are trying to calculate the number of seats a state should be apportioned, the state’s standard quota. So  So  would refer to seats allocated to a particular state, while  would refer to the state population. This means that the ratio of  to  is the ratio of the total population to house size, which is the standard divisor. So in apportionment terms, we have the following formula.

### Check Your Understanding

### Key Terms

1. apportion
2. apportionment problem
3. proportional
4. part-to-part ratio
5. representative democracies
6. standard divisor
7. states
8. seats
9. house size
10. state population
11. total population
12. standard quota

### Key Concepts

1. The apportionment problem is how to fairly divide and distribute available resources to recipients in whole, not fractional, parts.
2. To distribute the seats in the U.S. House of Representatives fairly to each state, calculations are based on state population, total population, and house size, or the total number of seats to be apportioned.
3. The standard divisor is the ratio of the total population to the house size, and the standard quota is the number of seats that each state should receive.

### Formulas

Let  be a particular item and  another such that there is a constant ratio of  to .
1.  and
2.
3.

### Videos

1. What Is a Ratio?
2. What Are the Different Types of Ratios?
3. Math Antics – Rounding