Classification of Quadrilaterals Classification of Quadrilaterals Quadrilateral is a geometric shape that consists of four points (vertices) sequentially joined by straight line segments (sides). We find the etymology of the word in S. Schwartzman's The Words of Mathematics : quadrilateral (noun, adjective): the first element is from Latin quadri- "four" from the Indo-European root kwetwer- "four." The second element is from Latin latus, stem later-, "side," of unknown prior origin. A quadrilateral is a four-sided polygon. The Latin term is a partial translation of Greek tetragon, literally "four angles," since a closed figure with four angles also has four sides. Although we use words like pentagon and polygon, the term quadrilateral has completely replaced tetragon. The seldom used term quadrangle has exactly the same meaning as quadrilateral, however the two related terms -- complete quadrangle and complete quadrilateral -- describe essentially different configurations. A quadrilateral may be convex or concave (see the diagram below.) A quadrilateral that is concave has an angle exceeding 180o.

In either case, the quadrilateral is simple, which means that the four sides of the quadrilateral only meet at the vertices, two at a time. So that two non-adjacent sides do not cross. A quadrilateral that is not simple is also known as self-intersecting to indicate that a pair of his non-adjacent sides intersect. The point of intersection of the sides is not considered a vertex of the quadrilateral. The shapes of elementary geometry are invariably convex. Starting with the most regular quadrilateral, namely, the square, we shall define other shapes by relaxing its properties. A square is a quadrilateral with all sides equal and all angles also equal. Angles in any quadrilateral add up to 360°. It follows that, in a square, all angles measure 90°. An equiangular quadrilateral, i.e. the one with all angles equal is a rectangle. All angles of a rectangle equal 90°. An equilateral quadrilateral, i.e. the one with all sides equal, is a rhombus. In a square, rectangle, or rhombus, the opposite side lines are parallel. A quadrilateral with the opposite side lines parallel is known as a parallelogram.

If only one pair of opposite sides is required to be parallel, the shape is a trapezoid. A trapezoid, in which the non-parallel sides are equal in length, is called isosceles. A quadrilateral with two separate pairs of equal adjacent sides is commonly called a kite. However, if the kite is concave, a dart is a more appropriate term. Kite and dart are examples of orthodiagonal quadrilaterals, i.e. quadrilaterals with perpendicular diagonals. A square and a rhombus are also particular cases of this class. The four vertices of a quadrilateral may be concyclic, i.e., lie on the same circle. In this case, the quadrilateral is known as circumscritptible or, simpler, cyclic. If a quadrilateral admits an incircle that touches all four of its sides (or more generally, side lines), it is known as inscriptible. A quadrilateral, both cyclic and inscriptible, is bicentric . The diagram below (which is a modification of one from wikipedia.org ) summarize the relationship between various kinds of quadrilaterals: The applet below illustrates the properties of various quadrilaterals.

In the applet, one can drag the vertices and the sides of the quadrilateral. You can display its diagonals, angle bisectors and the perpendicular bisectors of its sides. With these props, it's a simple matter to observe every single kind of quadrilateral, with a possible exception of bicentric. Which, too, is not overly difficult if you first get an isosceles trapezoid.) This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at download and install Java VM and enjoy the applet. What if applet does not run? As in the classification of triangles , the definitions may be either inclusive or exclusive. For example, trapezoid may be defined inclusively as a quadrilateral with a pair of parallel opposite sides, or exclusively as a quadrilateral with exactly one such pair. In the former case, parallelogram is a trapezoid, in the latter, it is not. Similarly, a square may or may not be a rectangle or a rhombus. My preference is with the inclusive approach.

For, I'd like to think of a square as a rhombus with right angles, or as a rectangle with all four sides equal. Here is a list of all the properties of quadrilaterals that we have mentioned along with the classes of the quadrilaterals that possess those properties: Property Rectangle, Square Orthodiagonal or inscriptible parallelogram is a rhombus; cyclic parallelogram is a rectangle. In particular, a parallelogram with equal diagonals is necessarily a rectangle. And not to forget, every simple quadrilateral tiles the plane . A simple quadrilateral with two pairs of equal opposite angles is a parallelogram. (Because then the opposite sides are parallel.) A simple quadrilateral with two pairs of equal opposite sides is a parallelogram. (Because of SSS when you draw one of the diagonals.) There is a simple quadrilateral with two pairs of equal sides: a kite (or a dart). It does have a pair of opposite equal angles. Nathan Bowler suggested a general construction of a quadrilateral with a pair of equal opposite sides and a pair of equal opposite angles which is not necessarily a parallelogram (there is a dynamic illustration ): Let ABC be isosceles with AB = AC.

Pick D on BC. Let C' be the reflection of C in the perpendicular bisector of AD. ABDC' has two opposite sides the same length and two opposite angles equal but is not a parallelogram if D isn't the midpoint of AB. This construction gives all such quadrilaterals. For an isosceles trapezoid ABCD with AB = CD, the quadrilateral ABDC has a pair of equal opposite sides and two pairs of equal opposite angles. Quadrilaterals Math Help > Geometry > Quadrilaterals Classification of quadrilaterals Depiction of a hierarchy of quadrilaterals, in which the green arrows represent additional properties. For example, a parallelogram becomes a rhombus by making all four sides the same length. Similar charts can be found in Mathwords , Wikipedia , and Dr.Math . Figure Definition, followed by additional facts (theorems) Internet references A closed plane figure consisting of four line segments. A "crossed" (or "complex") quadrilateral has one pair of intersecting sides. A "concave" quadrilateral, pictured here, has one interior angle greater than 180Â°. The other possibility, described next, is "convex".

Bretschneider's Formula gives the area of a quadrilateral with sides of length a, b, c, d and opposite interior angles A and C. The sum of the interior angles of a non-crossed quadrilateral 360. In a crossed quadrilateral, the sum of the interior angles on one side of the crossing equals the sum of the interior angles on the other side of the crossing. Mathwords , Mathworld , Dr.Math , Wikipedia , Geometry Atlas , Math.com , Math is fun "convex" means every line segment connecting interior points is entirely contained within the interior. Theorems: A case of Ramsey's Theorem tells us: Given any five points in a plane with no three collinear, four are the vertices of a convex quadrilateral. Mathwords , Dr.Math A quadrilateral with two pairs of adjacent equal sides. (In some text, a kite need not be convex; in others concave kites are termed a "dart" or "arrowhead".) Theorems: One set of opposite angles is equal, and that one diagonal perpendicularly bisects the other. The bisecting diagonal forms an axis of symmetry, dividing the kite into two congruent triangles.

The bisected diagonal divides the kite into two isosceles triangles. A convex kite is tangential (inscriptable). A quadrilateral has bilateral symmetry iff it is either a kite or an isosceles trapezoid. Mathwords , Mathworld , Dr.Math , Wikipedia , Geometry Atlas A convex quadrilateral whose four vertices lie on a circumscribed circle. Theorems: In a cyclic quadrilateral, opposite angles are supplementary. The area of a cyclic quadrilateral is the maximum possible for any quadrilateral with the given side lengths. Brahmagupta's formula gives the area of a cyclic quadrilateral, using only the side lengths. Cyclic quadrilaterals that are also inscriptable are called "bicentric". Mathwords , Mathworld , Dr.Math , Wikipedia , Geometry Atlas A kite which is also cyclic. Theorems: A cyclic kite has a pair of opposite right angles. Being a kite, it has a pair of congruent angles. Being cyclic, these congruent angles must also be supplementary, so they are right angles. A trapezoid with two opposite sides parallel, the two other sides are of equal length.

This implies that the two ends of each parallel side have equal angles, and that the diagonals are of equal length. A quadrilateral has bilateral symmetry iff it is either a kite or an isosceles trapezoid. Mathwords , Mathworld , Wikipedia A convex quadrilateral in which both pairs of opposite sides are parallel. This implies that opposite sides are of equal length, opposite angles are equal, and the diagonals bisect each other. Each diagonal bisects the parallelogram into two congruent triangles. Euclid showed if lines parallel to the sides are drawn through any point on a diagonal of a parallelogram, then the parallelograms not containing segments of that diagonal are equal in area (and conversely). Varignon's Theorem: A parallelogram is formed by joining the midpoints of adjacent sides of a quadrilateral. The center of Varignon's Parallelogram is the centroid of the vertices of the quadrilateral. Wittenbauer's Theorem: A parallelogram is formed by dividing the sides of a quadrilateral into three equal parts, and connecting and extending adjacent points on either side of each vertex. The center of Wittenbauer's parallelogram is the quadrilateral's centroid.

Mathworld , Wikipedia , Geometry Atlas A convex quadrilateral with all four sides of equal length. This implies that opposite sides are parallel, opposite angles are equal, and the diagonals perpendicularly bisect each other. Its area is given by A=bh, where b is the base, and h is the height, or perpendicular distance between opposite sides. A rhombus is tangential (inscriptable). Mathwords , Mathworld , Wikipedia , Geometry Atlas A convex quadrilateral with four sides of equal length, and four right angles. This implies that opposite sides are parallel, and that the diagonals perpendicularly bisect each other and are of equal length. Each diagonal bisects each pair of opposite angles. Tangential quadrilateral Inscriptable quadrilateral A convex quadrilateral in which a circle can be inscribed, tangent to all four sides. Theorems: The four angle bisectors meet (at the center of the inscribed circle) iff the figure is tangential. Pairs of opposites sides sum to the same number, which is the semiperimeter of the quadrilateral.

Cyclic inscriptable quadrilaterals are also called "bicentric" Wikipedia A quadrilateral with just one pair of congruent opposite sides, and just one pair of congruent opposite angles is not necessarily a parallelogram. The reason, as explained in Dr. Math, can be seen by drawing the shorter diagonal in the figure to the left, which divides the figure into two non-congruent triangles which nonetheless have congruent side-side-angle (SSA). Quadrilaterals | Free Math Worksheets Quadrilaterals Quadrilaterals are plane figures (polygons) that have four sides (called edges) and four angles (corners or vertices). They are sometimes also called quadrangles or tetragons. Broadly, they can be divided into three types known as topological types. These types are: convex, concave and crossed quadrilaterals. Convex and concave quadrangles are also considered simple quadrilaterals and as such the sum of their interior angles adds up to 360 degrees. This is a very useful piece of information as you are going to use it very often when calculating missing angles and the procedure is basically the same as it is with triangles . Types of quadrilaterals There are several different types of convex quadrilaterals.

These are the ones you will encounter most often: - Rhombus – all of its sides are the same length. It is often called a diamond or a lozenge, although the latter name refers to a specific rhombus that has a 45 degree angle. - Rhomboid – the length of its adjacent sides is unequal and its angles are oblique. - Rectangle – a quadrangle with four right angles. - Square – a rectangle with four equal sides. - Oblong – a non-square rectangle. In other words, every rectangle with unequal adjacent sides is considered an oblong. - Trapezoid – called trapezium in the UK, it is a convex tetragon with one pair of parallel sides. - Isosceles trapezoid – a special type of trapezoid. Its base angles have the same measure and the other pair of opposite sides is the same length. The square, rectangle, rhombus and rhomboid are considered parallelograms – tetragons with two pairs of parallel sides. Finding the area of quadrilaterals As it is with most plane figures, the area of some quadrilaterals is easier to calculate than other.

Quadrilaterals that demand the least amount of effort in order to calculate their areas are squares. The only thing you have to do to find the area of a square is to multiply the length of one of its sides with itself (does not matter which one – they are all the same length). The second easiest quadrilateral to work with is the rectangle. Its area can be calculated by simply multiplying the length of side a (which is one length) with the length of side b (which is a different length). Since there are two pairs of sides with the same length in a rectangle, this is a very simple task. The procedure for calculating the area of other quadrilaterals will be explained via examples. Example 1: Calculate the area of a parallelogram (K) whose height is 90mm and whose base is 9 cm long. Again, this is easier than it looks. Because a parallelogram consists of two pairs of equally long parallel sides, its area can be calculated using the same formula that is used in calculating the area of a rectangle. The only difference is that instead of multiplying two sides of the quadrilateral, we multiply the length of its base with its height (remember the triangle ).

The one thing you should be mindful of is the use of the units of measurement. Make sure they are consistent. If they are not, convert them to the most convenient unit. So, in this case we get: K = b * h K = 9 * 9 K = 81 cm2 We can now see that the area of that particular parallelogram is 81 cm2. We converted 90 mm into 9 cm (1cm equals 10mm) for the purpose of consistency and convenience (smaller numbers ). Now, let us solve another example. Example 2: Calculate the area of this trapezoid. Its height is 980 cm, the smaller base (a) is 440 cm long and the length of its larger base (b) is 13.8 m. First, let us convert the centimeters into meters for easier calculation. Since one meter equals 100 cm, the height of the trapezoid is 9.8 m and the length of the smaller base is 4.4 meters. Now, we have to find a way to calculate the area of this trapezoid.

It consists of one rectangle and two (in this case –different) triangles. You will maybe see it better if we rearrange it a bit to look like this: Now you see that this trapezoid consists of a rectangle whose sides are 4.4 and 9.8 meters long and two triangles whose measures we do not fully know yet. But if you look at them closely, you will notice that you can encase them in a rectangle whose sides are 9.4 (we subtract the length of the shorter base from the length of the longer base) and 9.8 meters (the height). Since these two triangles are right triangles, their combined area is half the area of the rectangle we mentally drew around them.

So, the size of the area of this trapezoid would be: K = ((a + b) / 2) * h K = ((4.4 + 13.8) / 2) * 9.8 K = (18.2 / 2) * 9.8 K = 9.1 * 9.8 K = 89.18 m2 This is the basic information you need to wrap your head around these very common plane figures. If you wish to practice what you learned about quadrilaterals, please feel free to use the math worksheets below. Quadrilaterals exams for teachers Quadrilateral | MathCaptain.com Quadrilateral Quadrilateral Get a Tutor When we think of a room, its walls and floors, a news paper, a note book, the computer screen, the surfaces of a rectangular box etc, we see flat surfaces which are rectangular. The rectangular surface is the one which has four sided and the sides are straight. So we can see that the rectangular surfaces are those which are bounded by four sides. All four sided figures are not rectangles. The four sided figures called quadrilaterals are classified into different types according to the sides and the angles.

In this section let us see the definition of quadrilaterals and the other types of quadrilaterals and their properties. What is a Quadrilateral? Quadrilateral is a four sided polygon. It is a closed figure bounded by four sides. Example: Floor of a room, walls of a room, ceiling of a room, computer screen, The surface of a note book. Types of Quadrilaterals: The quadrilaterals are classified into different names according to the relationship of their sides and the angles. The different types of Quadrilaterals are, Parallelogram, rhombus, rectangle, square, trapezoid, concave and convex quadrilaterals. 1. Parallelogram: It is a quadrilateral whose pair of opposite sides are parallel. 2. Rhombus: It is a parallelogram whose sides are of congruent. 3. Rectangle: It is a parallelogram whose interior angles measure 90o . 4. Square: It is a parallelogram whose sides are congruent and the interior angles measure 90o . 5. Trapezoid: It is a quadrilateral whose one pair of opposite sides are parallel. 6.

Kite: It is a quadrilateral which has two distinct pairs of adjacent sides are congruent. Regular Quadrilateral: Regular Quadrilaterals are those in which the sides are of equal length and interior angles are equal. Square is a regular quadrilateral whose sides are of equal measure and the interior angles measure 90o . Irregular Quadrilateral: The quadrilaterals which are not regular, are called irregular quadrilaterals. In an irregular quadrilateral, all sides are not equal and the interior angles are not of same measure. Concave Quadrilateral: Concave quadrilaterals, are those in which one of its interior angle measure more than 180o (reflex angle). The above diagram shows a concave quadrilateral. Convex Quadrilateral: A quadrilateral whose interior angles measure less than 1800 ( acute or obtuse ) is called a convex quadrilateral. Square, Rectangle, Parallelogram, Rhombus, Trapezoid are all convex quadrilateral. Properties of Quadrilaterals: 1. Properties of a Parallelograms: a. Opposite sides are parallel. b. Opposite sides are congruent. c. Opposite angles are congruent. d.

Pair of consecutive angles are supplementary. e. Diagonals bisect each other. 2. Properties of a Rhombus: a. Opposite sides are parallel. b. All the four sides congruent. c. Opposite angles are congruent. d. Pair of consecutive angles are supplementary. e. Diagonals bisect each other. 3. Properties of a Rectangle: a. Opposite sides are parallel. b. Opposite sides are congruent. c. Each of interior angle measure 90o . d. Diagonals are congruent. e. Diagonals bisect each other. 4. Properties of a Square: a. Opposite sides are parallel. b. All the four sides are congruent. c. Each of interior angle measure 90o. d. Diagonals are congruent. e. Diagonals bisect each other. 5. Properties of a Trapezoid: a. It has one pair of opposite sides parallel. b. The interior angles on the same side of the non-parallel lines are supplementary. 6. Properties of Isosceles Trapezoid: a. It has one pair of opposite sides parallel. b. The non parallel sides are congruent. c.

The lower base angles are congruent. d. The upper base angles are congruent. e. Diagonals are congruent. f. Pair of a lower base angle and an upper base angle is supplementary. Similar Quadrilaterals: The pair of quadrilaterals are said to be similar if their corresponding interior angles are equal and the corresponding sides are proportional. The following diagram shows pairs of quadrilaterals which are similar. Cyclic Quadrilaterals: Quadrilaterals whose vertices lie on the circumference of a circle are called cyclic quadrilaterals. The following diagram shows the cyclic quadrilaterals. The Properties of Cyclic Quadrilaterals: 1. Opposite angles of a cyclic quadrilateral is equal to 180o . 2. Exterior angles of a cyclic Quadrilateral is equal to the interior opposite angle. Sum of all the angles of a quadrilateral is equal to 360o Solved Example Question: The interior angles of quadrilateral are in the ratio 3 : 5 : 7 : 9, Find the measure of each angle of the quadrilateral.

Solution: Area and Perimeter of Quadrilateral Area of Quadrilateral: Area of a quadrilateral is the sum of the areas of the two triangles formed by one of its diagonal. If the perpendicular drawn to one of the diagonals from the opposite vertices are given by h1 and h2 . Then the area of each triangle = $\frac{1}{2}$ x diagonal x h1 and $\frac{1}{2}$ x diagonal x h2 Therefore, sum of the areas of the two triangles = $\frac{1}{2}$ x diagonal x h1 + $\frac{1}{2}$ x diagonal x h2 = $\frac{1}{2}$ x diagonal x (h1 + h2) Areas of different types of quadrilaterals: 1. Area of a parallelogram = base x Height = bh 2. Area of a rectangle = length x width = l w 3. Area of a square = side x side = s2 4. Area of a rhombus = $\frac{1}{2}$ Diagonal 1 x Diagonal 2 5.

Area of a trapezoid = $\frac{1}{2}$ x Height x (sum of the parallel sides) = $\frac{1}{2}$ x h x ( a + b) , where h is the perpendicular distances between the parallel sides a and b. Perimeter of a quadrilateral: Perimeter of a quadrilateral is the sum of all the sides. Perimeter of different types of quadrilaterals: 1. Perimeter of a parallelogram = 2 (sum of the two adjacent sides) 2. Perimeter of a rectangle = 2 (length + width) 3. Perimeter of a square = 4 x side = 4 s 4. Perimeter of a rhombus = 4 x side = 4 s 5. Perimeter of a trapezoid = sum of the parallel sides + sum of non-parallel sides. Prove that the opposite angles of a parallelogram are congruent. Given : In the above diagram, ABCD is a parallelogram. To Prove : $\angle A$ = $\angle C$ and $\angle B$ = $\angle D$ Construction: Join AC.

Diagonals in Quadrilaterals Diagonals in Quadrilaterals Main Theorems The diagonals of a quadrilateral can determine whether it is a parallelogram, a rectangle, a rhombus, etc.. We will list and prove the main theorems here. Theorem 1: If the diagonals of a quadrilateral bisect each other then the quadrilateral is a parallelogram. Proof: Let the quadrilateral be ABCD with diagonals AC and BD intersecting at P: Then AP = PC and DP = PB since the diagonals bisect each other. Now consider triangles APD and CPB. The vertical angles APD and CPB are equal, and we have pairs of sides that are equal (AP = PC and DP = PB), so these triangles are congruent by SAS. As a consequence, the corresponding angles, DAP and BCP are equal. But these are alternate interior angles for lines AD and BC with transversal AC. So this proves side AD is parallel to side BC. Similarly, if we consider triangles APB and CPD, they are congruent by the same reasoning, so corresponding angles BAP and DCP are equal.

These are alternate interior angles for lines AB and DC, so those lines are also parallel. Therefore ABCD is a parallelogram, by definition. Theorem 2: If the diagonals of a quadrilateral bisect each other and have the same length, then the quadrilateral is a rectangle. Proof: Since the diagonals bisect each other, we already know (from Theorem 1) that it is a parallelogram, so all we need to prove is that the angles at the vertices are right angles. Again let the quadrilateral be ABCD with diagonals AC and BD intersecting at P. Since the diagonals bisect each other, P is the midpoint of both diagonals. That is, AP = PB and CP = PD. But the diagonals are also of the same length, so AP + PB = CP + PD, and by substitution this gives us AP + AP = CP + CP, or 2AP= 2CP. That is, AP = CP. Likewise, PB = PD. Consequently, all 4 triangles APD, APB, CPD, and BPC are isosceles.

So angles PAD and PDA are congruent, angles PBC and PCB are congruent, angles PAB and PBA are congruent, and angles PDC and PCD are congruent. We already saw in the proof of Theorem 1 that triangles APD and CPB are congruent, as are triangles APB and CPD. By angle addition, it follows that the 4 angles of the quadrilateral (angles ABC, BCD, CDA, and DAB) are all equal. But the angles of a quadrilateral add to 360o, and therefore each of these 4 angles must be 90o. Theorem 3: If the diagonals of a quadrilateral bisect each other and are perpendicular then the quadrilateral is a rhombus. Proof: Again let the quadrilateral be ABCD with diagonals AC and BD intersecting at P: Since they bisect each other and are perpendicular, triangles APB, BPC, CPD, and DPA are right triangles. They are all congruent by SAS.

For example, triangle APB is congruent to triangle CPB because they share a common side BD, sides AP and CP are congruent (since P is the midpoint of AC), and the included angles are both right angles. Since they are all congruent, their third sides (the hypotenuse of each) are congruent (CPCTC). A rhombus is a quadrilateral in which all four sides are congruent, so ABCD is a rhombus. Theorem 4: If the diagonals of a quadrilateral are perpendicular and one bisects the other, then the quadrilateral is a kite. That is, if the quadrilateral is ABCD with diagonals AC and BC intersecting at P, and if AP = CP, then AB = BC and AD = BD: Proof: In this case, triangles APB and CPB are congruent (by SAS), and triangles APD and CPD are congruent. Therefore AB = BC and AD = CD. Narrowing the Type When a quadrilateral is known to be of certain special type, then additional properties of the diagonals can narrow the type.

The following theorems demonstrate this: Theorem 5: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. Proof: Let the parallelogram be ABCD with congruent diagonals AC and BD. Consider the overlapping triangles ADC and BCD. Since opposite sides of a parallelogram are congruent, AD = BC. Since the diagonals of the parallelogram are congruent, AC = BD, and the overlapping triangles have a common side, DC. Therefore they are congruent by SSS. So angle ADC and BCD are congruent. But these are same-side interior angles for parallel lines AD and BC with transversal DC. Since same-side interior angles add to 180o, each must be 90o, so the parallelogram is a rectangle. This theorem is often used by carpenters to check a door or window to see if it is really rectangular. First the carpenter measures the opposite sides. If they are the same length, then he measures the diagonals. If they too are the same length, then he knows the angles are right angles.

Theorem 6: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. Proof: Let the parallelogram be ABCD with perpendicular diagonals AC and BD intersecting at P: The diagonals of a parallelogram bisect each other, so triangles APB and CPB are congruent by SAS. Therefore the corresponding parts, sides AB and CB are congruent. Likewise, triangles BPC and DPC are congruent, so sides BC and DC are congruent, and similarly sides AD and CD are congruent. So all 4 sides are congruent, which makes the parallelogram a rhombus. There are other theorems which could be stated, but the main ideas revolve around congruent triangles formed by the diagonals and sides of the quadrilateral. Area of Rhombus | Area of a Rhombus with Diagonals | MathCaptain.com Area of a Rhombus with Diagonals When the length of the diagonals is known, area of rhombus is half the product of the diagonals.

Area of rhombus = $\frac{D_{1}D_{2}}{2}$where $D_{1}$ : Length of first diagonal and $D_{2}$ : Length of second diagonal. Finding the Area of a Rhombus Solved Examples Question 1: Diagonals of the rhombus are 25 cm and 12 cm find the area of the rhombus. Solution: As the diagonals are given in the problem we use the below formula to find the area of the rhombus. Area of rhombus = $\frac{D_{1}D_{2}}{2}$ Plug in the given values $D_{1}$ = 25 cm and $D_{2}$ = 12 cm. Area of rhombus = $\frac{25 \times 12}{2}$ $\Rightarrow$ $\frac{300}{2}$ = 150 Therefore area of the rhombus = 150 cm$^{2}$ Question 2: Find the area of the rhombus, if perimeter is 100 cm and the length of the diagonal is 16 cm.

Solution: Tessellations by Squares, Rectangles and other Polygons - EscherMath Tessellations by Squares, Rectangles and other Polygons From EscherMath K-12: Materials at high school level. Oberkapfenberg Castle, Styria, Austria. 7 Notes Some Basic Tessellations Recall that a polygon is just a simple geometric shape. The polygons we will be talking about are squares, rectangles, parallelograms, the rhombus and maybe a couple of other ones. If you haven't thought about polygons for a couple of weeks it may be useful to quickly sneek a peek at some of the examples we talked about earlier: Squares, Rectangles, Parallelograms and Other Polygons In this section we will be looking at the following: Question: Which polygons tile the plane? Another way of saying this is: which polygons can be used to tile say a floor or a wall? This means we are looking for shapes that fit together nicely, without any gaps or overlaps to create a pattern. Some people call these patterns tilings, while others call them tessellations. Both words are correct. We may use them both in this text. The most common and simplest tessellation uses a square.

You may not have thought about it, but you will ahve seen titlings by squares before. A lot of bathrooms have square tiles on the floor. A lot of classsrooms will have squares on the floor and there may even be squares in the ceiling. Squares easily form horizontal strips: Stacks of these strips cover a rectangular region and the pattern can clearly be extended to cover the entire plane. This easily gives us the result that: All squares tessellate. The same technique works with parallelograms. You can put parallelograms side by side and create these strips. If you stack the trips you will have a tiling by parallelograms, and so: All parallelograms tessellate. Looking for other tessellating polygons is a complex problem, so we will organize the question by the number of sides in the polygon. The simplest polygons have three sides, so we begin with triangles: All triangles tessellate. To see this, take an arbitrary triangle and rotate it about the midpoint of one of its sides. The resulting parallelogram tessellates: The picture works because all three corners (A, B, and C) of the triangle come together to make a 180° angle - a straight line.

This property of triangles will be the foundation of our study of polygon tessellations, so we state it here: The sum of angles of any triangle is 180°. Moving up from triangles, we turn to four sided polygons, the quadrilaterals. Before continuing, try the Quadrilateral Tessellation Exploration . Tessellations by Quadrilaterals Recall that a quadrilateral is a polygon with four sides. The sum of angles in any quadrilateral is 360° To prove, divide a quadrilateral into two triangles as shown: Since the angle sum of any triangle is 180°, and there are two triangles, the angle sum of the quadrilateral is 180° + 180° = 360°. Taking a little more care with the argument, we have: and . This division into triangles does not calculate the angle sum of the quadrilateral. The point of all the letters is that the angles of the triangles make the angles of the quadrilateral, which would not work if the quadrilateral was divided as shown on the right. We now turn to the main result of this section: All quadrilaterals tessellate. Begin with an arbitrary quadrilateral ABCD.

Rotate by 180° about the midpoint of one of its sides, and then repeat using the midpoints of other sides to build up a tessellation. The angles around each vertex are exactly the four angles of the original quadrilateral. Since the angle sum of the quadrilateral is 360°, the angles close up, the pattern has no gaps or overlaps, and the quadrilateral tessellates. Recall from Fundamental Concepts that a convex shape has no dents. All triangles are convex, but there are non-convex quadrilaterals. The technique for tessellating with quadrilaterals works just as well for non-convex quadrilaterals: It is worth noting that the general quadrilateral tessellation results in a wallpaper pattern with p2 symmetry group. Tessellations by Convex Polygons Every shape of triangle can be used to tessellate the plane. Every shape of quadrilateral can be used to tessellate the plane. In both cases, the angle sum of the shape plays a key role.

Since triangles have angle sum 180° and quadrilaterals have angle sum 360°, copies of one tile can fill out the 360° surrounding a vertex of the tessellation. The next simplest shape after the three and four sided polygon is the five sided polygon: the pentagon. The angle sum of any pentagon is 540°, because we can divide the pentagon into three triangles: Since each triangle has angle sum 180° the angle sum of the pentagon is 180° + 180° + 180° = 540°. Rather than repeat the angle sum calculation for every possible number of sides, we look for a pattern. The angle sum of a triangle (3-gon) is 180°, the angle sum of a quadrilateral (4-gon) is 2x180°, and the angle sum of a pentagon is 3x180°. A general polygon with sides can be cut into triangles and so we have: The sum of the angles of an -gon is .

Unlike the triangle and quadrilateral case, the pentagon's angle sum of 540° is not helpful when trying to fit a bunch of pentagons around a vertex. In fact, there are pentagons which do not tessellate the plane. For example, the regular pentagon has five equal angles summing to 540°, so each angle of the regular pentagon is . Attempting to fit regular polygons together leads to one of the two pictures below: Both situations have wedge shaped gaps that are too narrow to fit another regular pentagon. Thus, not every pentagon tessellates. On the other hand, some pentagons do tessellate, for example this house shaped pentagon: The house pentagon has two right angles. Because those two angles sum to 180° they can fit along a line, and the other three angles sum to 360° (= 540° - 180°) and fit around a vertex. Thus, some pentagons tessellate and some do not. The situation is the same for hexagons, but for polygons with more than six sides there is the following: No convex polygon with seven or more sides can tessellate.

This remarkable fact is difficult to prove, but just within the scope of this book. However, the proof must wait until we develop a counting formula called the Euler characteristic, which will arise in our chapter on Non-Euclidean Geometry . Nobody has seriously attempted to classify non-convex polygons which tessellate, because the list is quite likely to be too long and messy to describe by hand. However, there has been quite a lot of work towards classifying convex polygons which tessellate. Because we understand triangles and quadrilaterals, and know that above six sides there is no hope, the classification of convex polygons which tessellate comes down to two questions: Which convex pentagons tessellate? Which convex hexagons tessellate? Question 2 was completely answered in 1918 by K. Reinhardt. [1] Reinhardt showed that there are only three types of convex hexagons which tessellate: File:Convex-hexagons.svg Reinhardt also addressed Question 1 and gave five types of pentagon which tessellate. In 1968, R. Kershner [2] found three new types, and claimed a proof that the eight known types were the complete list.

A 1975 article by Martin Gardner [3] in Scientific American popularized the topic, and led to a surprising turn of events. In fact Kershner's "proof" was incorrect. After reading the Scientific American article, a computer scientist, Richard James III, found a ninth type of convex pentagon that tessellates. Not long after that, Marjorie Rice , a San Diego homemaker with only a high school mathematics background, discovered four more types, and then a German mathematics student, Rolf Stein, discovered a fourteenth type in 1985. Since 1985, no new types have been discovered, and many mathematicians believe that the list is finally complete. However, there is no well accepted proof of the classification, so it remains possible that there is a fifteenth or even many more types of convex pentagons that tessellate. Today, question 1 is an open problem, a problem whose solution is unknown. Summary of Polygon Tessellations Regular tessellation A tessellation using one regular polygon tile, arranged so that edges match up.

Corners of the tiles need to fit together around a point, which means the corner angle of the regular polygon must evenly divide 360°. Since , there is a regular tessellation using six triangles around each vertex. Since , there is a regular tessellation using four squares around each vertex. And since there is a regular tessellation using three hexagons around each vertex. We have already seen that the regular pentagon does not tessellate. A regular polygon with more than six sides has a corner angle larger than 120° (which is 360°/3) and smaller than 180° (which is 360°/2) so it cannot evenly divide 360°. We conclude: There are three regular tessellations of the plane: by triangles, by squares, by hexagons. A major goal of this book is to classify all possible regular tessellations. Apparently, the list of three regular tessellations of the plane is the complete answer. However, these three regular tessellations fit nicely into a much richer picture that only appears later when we study Non-Euclidean Geometry .

Tessellations using different kinds of regular polygon tiles are fascinating, and lend themselves to puzzles, games, and certainly tile flooring. Try the Pattern Block Exploration . Archimedean tessellations - * Optional An Archimedean tessellation (also known as a semi-regular tessellation) is a tessellation made from more that one type of regular polygon so that the same polygons surround each vertex. Archimedean Tessellations (4,6,12) (3,3,3,3,6) We can use some notation to clarify the requirement that the vertex configuration be the same at every vertex. We can list the types of polygons as they come together at the vertex. For instance in the top row we see on the left a semi-regular tessellation with at every vertex a (3,6,3,6) configuration. We see a 3-gon, a 6-gon, a 3-gon and a 6-gon. The other tessellations on the top row have a (3,4,6,4), a (3,12,12), and a (3,3,3,4,4) configuration.

These configurations are unique up to cyclic reordering (and possibly reversing the order). For example (3,12,12) can also be written as (12,12,3) or (12,3,12). In the bottom row we have (4,8,8), (3,3,4,3,4), (4,6,12) and (3,3,3,3,6) configurations. Recall that creating a tessellation requires the angle sums of the polygons to add up to 360° around a vertex. We know from a previous section that the angle of a regular n-gon is . If we have a collection of n-gons: , then the angles of those polygons need to add up to 360°: Dividing both sides by 180° gives us the following equation: . For example, suppose that , then we have: . This means that 3 triangles and 2 squares will give us a vertex type. In this case we can arrange these polygons around the vertex in two different ways: (3,3,3,4,4) and (3,3,4,3,4).

Both of these will give rise to a semi-regular tessellation. Recall that a semi-regular tessellation satisfies the following requirements: All the tiles are regular polygons It is a tessellation, hence no gaps or overlaps All the vertices are of the same type. There are only 21 combinations of regular polygons that will fit around a vertex. And of these 21 there are there are only 11 that will actually extend to a tessellation. Below are the different vertex types. An asterisk indicates that this vertex type cannot be extended to a tessellation. type

Question: A simple quadrilateral whose sides are all of the same length is known as a what?
Answer: RHOMBUS