SparkNotes: SAT Subject Test: Math Level 1: Solids Produced by Rotating Polygons Solids Produced by Rotating Polygons Solids Produced by Rotating Polygons Another type of Math IC question that you may come across involves a solid produced by the rotation of a polygon. The best way to explain how this type of problem works is to provide a sample question: What is the surface area of the geometric solid produced by the triangle below when it is rotated 360 degrees about the axis AB? When this triangle is rotated about AB, a cone is formed. To solve the problem, the first thing you should do is sketch the cone that the triangle will form. The question asks you to figure out the surface area of the cone. The formula for surface area is πr2 + πrl, which means you need to know the lateral height of the cone and the radius of the circle. If you’ve drawn your cone correctly, you should see that the lateral height is equal to the hypotenuse of the triangle. The radius of the circle is equal to side BC of the triangle. You can easily calculate the length of BC since the triangle is a 30-60-90 triangle.

If the hypotenuse is 2, then BC, being the side opposite the 30º angle, must be 1. Now plug both values of l and r into the surface area formula and then simplify: Common Rotations You don’t need to learn any new techniques or formulas for problems that deal with rotating figures. You just have to be able to visualize the rotation as it’s described and be aware of which parts of the polygons become which parts of the geometric solid. Below is a summary of which polygons, when rotated a specific way, produce which solids. A rectangle rotated about its edge produces a cylinder. A semicircle rotated about its diameter produces a sphere. A right triangle rotated about one of its legs produces a cone. A rectangle rotated about a central axis (which must contain the midpoints of both of the sides that it intersects) produces a cylinder. A circle rotated about its diameter produces a sphere. An isosceles triangle rotated about its axis of symmetry (the altitude from the vertex of the non-congruent angle) produces a cone. Mathematics Benchmarks, Grades K-12 G.K.1 Create, explore, and describe shapes. a.

Identify common shapes such as rectangle, circle, triangle, and square. Draw a variety of triangles (equilateral, right, isosceles, scalene) in a variety of positions. Draw squares and rectangles of different proportions (tall, squat, square-like) both horizontally and vertically positioned. Recognize tipped squares and rectangles. Note: Squares are a type of rectangle. Note: Drawing of tipped rectangles is typically too hard for kindergarten. Describe attributes of common shapes (e.g., number of sides and corners). b. Use geometric tiles and blocks to assemble compound shapes. Assemble rectangles from two congruent right triangular tiles. Explore two-dimensional symmetry using matching tiles. c. Recognize and use words that describe spatial relationships such as above, below, inside, outside, touching, next to, far apart. Grade 1 G.1.1 Recognize, describe, and draw geometric figures. a. Identify and draw two-dimensional figures. Include trapezoids, equilateral triangles, isosceles triangle, parallelograms, quadrilaterals. Note: Be sure to include a robust variety of triangles as examples, especially ones that are very clearly not equilateral or isosceles.

Describe attributes of two-dimensional shapes (e.g., number of sides and corners). b. Identify and name three-dimensional figures. Include spheres, cones, prisms, pyramids, cubes, rectangular solids. Identify two-dimensional shapes as faces of three-dimensional figures. c. Sort geometric objects by shape and size. Recognize the attributes that determined a particular sorting of objects and use them to extend the sorting. Example: Various L-shaped figures constructed from cubes are sorted by the total number of cubes in each. Recognize this pattern, then sort additional figures to extend the pattern. Explore simultaneous independent attributes. Example: Sort triangular tiles according to the four combinations of two attributes, such as right angle and equal sides. G.1.2 Rotate, invert, and combine geometric tiles and solids. a. Describe and draw shapes resulting from rotations and flips of simple two-dimensional figures. Identify the same (congruent) two-dimensional shapes in various orientations and move one on top of the other to show that they are indeed identical. Extend sequences that show rotations of simple shapes. b. Identify symmetrical shapes created by rotation and reflection. c. Use geometric tiles and cubes to assemble and disassemble compound figures.

Count characteristic attributes (lines, faces, edges) before and after assembly. Examples: Add two right triangles to a trapezoid to make a rectangle; create a hexagon from six equilateral triangles; combine two pyramids to make a cube. Grade 2 G.2.1 Recognize, classify, and transform geometric figures in two and three dimensions. a. Identify, describe, and compare common geometric shapes in two and three dimensions. Define a general triangle and identify isosceles, equilateral, right, and obtuse special cases. Note: The goal of naming triangles is not the names themselves but to focus on important differences. Triangles (or quadrilaterals) are not all alike, and it is their differences that give them distinctive mathematical features. Identify various quadrilaterals (rectangles, trapezoids, parallelograms, squares) as well as pentagons and hexagons. Note: In this grade parallel is used informally and intuitively; it receives more careful treatment at a later grade.

Note: A square is a special kind of rectangle (since it has four sides and four right angles); a rectangle is a special kind of parallelogram (since it has four sides and two pairs of parallel sides); and a parallelogram is a special kind of trapezoid (since it has four sides and at least one pair of parallel sides); and a trapezoid is a polygon (since it is a figure formed of several straight sides). So for example, contrary to informal usage, in mathematics, a square is a trapezoid. Understand the terms perimeter and circumference. Note: The primary meaning of both terms is the outer boundary of a two-dimensional figure; circumference is used principally in reference to circles. A secondary meaning for both is the length of the outer boundary. Which meaning is intended needs to be determined from context. Distinguish circles from ovals; recognize the circumference, diameter, and radius of a circle. In three dimensions, identify spheres, cones, cylinders, and triangular, and rectangular prisms. b. Describe common geometric attributes of familiar plane and solid objects. Common geometric attributes include position, shape, size, and roundness, and numbers of corners, edges, and faces.

Distinguish between geometric attributes and other characteristics such as weight, color, or construction material. Distinguish between lines and curves and between flat and curved surfaces. c. Rotate, flip, and fold shapes to explore the effect of transformations. Use paper folding to find lines of symmetry. Recognize congruent shapes. Identify shapes that have been moved (flipped, slid, rotated), enlarged, or reduced. G.2.2 Understand and interpret rectangular arrays as a model of multiplication. a. Create square cells from segments of the discrete number line used as sides of a rectangle. Match cells to discrete objects lined up in regular rows of the same length. b. Understand rectangular arrays as instances of repeated addition. Grade 3 G.3.1 Recognize basic elements of geometric figures and use them to describe shapes. a. Identify points, rays, line segments, lines, and planes in both mathematical and everyday settings. A line is a straight path traced by a moving point having no breadth nor end in either direction. Examples: Each figure on the left above represents a line; the arrows indicate that the lines keep going in the indicated directions without end. The number line with both positive and negative numbers is a line.

A part of a line that starts at one point and ends at another is called a line segment. Line segments are drawn without arrows on either end because line segments end at points. Examples: The figure in the center above is a line segment. The edges of a desk or door or piece of paper are everyday examples of line segments. Part of a line that starts at one point and goes on forever in one direction is called a ray. Examples: The figure above on the right is a ray. The positive number line (to the right of 0) is a ray. On the other hand, none of the four examples below are lines: Caution: Not all sources distinguish carefully among the terms line, segment, or ray, nor do all sources employ the convention of arrowheads in exactly the manner described above. Often context is the best guide to distinguish among these terms. Know that a plane is a flat surface without thickness that extends indefinitely in every direction. Examples: Everyday examples that illustrate a part of a plane are the flat surfaces of a floor, desk, windowpane, or book. Examples that are not part of a plane are the curved surfaces of a light bulb, a ball, or a tree. b.

Understand the meaning of parallel and perpendicular and use these terms to describe geometric figures. Lines and planes are called parallel if they do not meet no matter how far they are extended. Lines and planes are called perpendicular if the corners formed when they meet are equal. Identify parallel and perpendicular edges and surfaces in everyday settings (e.g., the classroom). Examples: The lines on the page of a notebook are parallel, as are the covers of a closed book. Corners of books, walls, and rectangular desks are perpendicular, as are the top and side edges of a chalk board and a wall and a floor in a classroom. The corner where two perpendicular lines meet is called a right angle. Note: The general concept of "angle" is developed later; here the term is used merely as the name for this specific and common configuration. Understand and use the terms vertical and horizontal. Recognize that vertical and horizontal lines or planes are perpendicular, but that perpendicular lines or planes are not necessarily vertical or horizontal. c. Use terms such as line, plane, ray, line segment, parallel, perpendicular, and right angle correctly to describe everyday and geometric figures. G.3.2 Identify and draw perpendicular and parallel lines and planes. a.

Draw perpendicular, parallel, and non-parallel line segments using rulers and squares. Recognize that lines that are parallel to perpendicular lines will themselves be perpendicular. Example: Fold a piece of paper in half from top to bottom, then fold it in half again from left to right. This will give two perpendicular fold lines and four right angles. Edges of a polygon are called parallel or perpendicular if they lie on parallel or perpendicular lines, respectively. Similarly, faces of a three-dimensional solid are called parallel or perpendicular if they lie in parallel or perpendicular planes, respectively. G.3.3 Explore and identify familiar two- and three-dimensional shapes. a. Describe and classify plane figures and solid shapes according to the number and shape of faces, edges, and vertices. Plane figures include circles, triangles, squares, rectangles, and other polygons; solid shapes include spheres, pyramids, cubes, and rectangular prisms. Recognize that the exact meaning of many geometric terms (e.g., rectangle, square, circle and triangle) depends on context: Sometimes they refer to the boundary of a region and sometimes to the region contained within the boundary. b. Know how to put shapes together and take them apart to form other shapes.

Examples: Two identical right triangles can be arranged to form a rectangle. Two identical cubes can be arranged to form a rectangular prism. See figures below. c. Identify edges, vertices (corners), perpendicular and parallel edges and right angles in two-dimensional shapes. Example: A rectangle has four pairs of perpendicular edges, two pairs of parallel edges and four right angles. d. Identify right angles, edges, vertices, perpendicular and parallel planes in three-dimensional shapes. G.3.4 Understand how to measure length, area, and volume. a. Understand that measurements of length, area, and volume are based on standard units. Fundamental units are: a unit interval of length 1 unit, a unit square whose sides have length 1 unit, and a unit cube whose sides have length 1 unit. The volume of a rectangular prism is the number of unit cubes required to fill it exactly (with no space left over). Note: The common childhood experience of pouring water or sand offers a direct representation of volume. The area of a rectangle is the number of unit squares required to pave the rectangle—that is, to cover it completely without any overlapping. Note: Area provides a critical venue for developing the conceptual underpinnings of multiplication.

The length of a line segment is the number of unit intervals that are required to cover the segment exactly with nothing left over. b. Know how to calculate the perimeter, area, and volume of shapes made from rectangles and rectangular prisms. The perimeter of a rectangle is the number of unit intervals that are required to enclose the rectangle. Measure and compare the areas of shapes using non-standard units (e.g., pieces in a set of pattern blocks). Recognize that the area of a rectangle is the product of the lengths of its base and height (A = b × h) and that the volume of a rectangular prism is the product of the lengths of its base, width, and height (V = b × w × h). Example: Build solids with unit cubes and use the formula for volume (V = bwh) to verify the count of unit cubes; make similar comparisons with rectilinear figures in the plane that are created from unit squares. Find the area of a complex figure by adding and subtracting areas. Compare rectangles of equal area and different perimeter and also rectangles of equal perimeter and different area. Measure surface area of solids by covering each face with copies of a unit square and then counting the total number of units.

Grade 4 G.4.1 Understand and use the definitions of angle, polygon, and circle. a. An angle in a plane is a region between two rays that have a common starting point. Note: According to this definition, a right angle (as determined by perpendicular rays) is indeed an angle. b. If angle A is contained in another angle B, then angle B is said to be bigger than angle A. The figures below illustrate how to determine whether an angle is larger than, smaller than, or close to a right angle. The angle is not contained in a right angle, so this tells us that it is larger than a right angle. The angle is contained in a right angle, so this tells us that it is smaller than a right angle. Note: When two rays come from the same point (see figure below), they divide the plane into two regions, giving two angles. Except where otherwise indicated, the angle determined by the two rays is defined, by convention, as the smaller region. Understand that shapes such as triangles, squares, and rectangles have angles. Note: Technically, polygons do not contain rays, which are required for the definition of angles. Their sides are line segments of finite length.

Nonetheless, if we imagine the sides extending indefinitely away from each corner, then each corner becomes an angle. Example: Describe the difference between the two figures below: Identify acute, obtuse and right angles. c. Know and use the basic properties of squares; rectangles; and isosceles, equilateral and right triangles. Identify scalene, acute, and obtuse triangles. Know how to mark squares, rectangles, and triangles appropriately: d. Know what a polygon is and be able to identify and draw some examples. A polygon is a figure that lies in a plane consisting of a finite number of line segments called edges (or sides) with the properties that (a) each edge is joined to exactly two other edges at the end points; edges do not meet each other except at end points; and the edges enclose a single region. Example: The figures below are not polygons: e. Know and use the basic properties of a circle. A circle is the set of points in a plane that are at a fixed distance from a given point. Know that a circle is not a polygon. Grade 5 G.5.1 Measure angles in degrees and solve related problems. a.

Understand the definition of degree and be able to measure angles in degrees. A degree is one part of the circumference of a circle of radius 1 unit (a unit circle) that is divided into 360 equal parts. The measure of an angle in degrees is defined to be the number of degrees of the arc of the unit circle, centered at the vertex of the angle, that is intercepted by the angle. The measure of an angle in degrees can also be interpreted as the amount of counter-clockwise turning from one ray to the other. Note: Earlier (in grade 4), the angle determined by two rays was defined to be the smaller of the two options. For consistency, therefore, when an angle is measured by the amount of turning necessary to rotate one ray into another, it is important to start with the particular ray that will produce an angle measure no greater than 180°. The symbol ° is an abbreviation for "degree" (e.g., 45 degrees = 45°). As a shorthand, angles are called equal if the measures of the angles are equal. b. Know and use the measures of common angles.

Recognize that angles on a straight line add up to 180° and that angles around a point add up to 360°. An angle of 180° is called a straight angle. A right angle is an angle of 90°. An acute angle is an angle of less than 90°, while an obtuse angle is an angle of more than 90°. Note: Since a pair of perpendicular lines divides the plane into 4 equal angles, the measure of a right angle is 360°/4 = 90°. c. Interpret and prepare circle graphs (pie charts). G.5.2 Know how to do basic constructions using a straightedge and compass. a. Basic constructions include (a) drop a perpendicular from a point to a line, (b) bisect an angle, (c) erect the perpendicular bisector of a line, and (d) construct a hexagon on a circle. Note: A straightedge is a physical representation of a line, not a ruler that is used for measuring.

The role of a straightedge in constructions is to draw lines through two points, just as the role of the compass is to draw a circle based on two points, the center and a point on the circumference. Note: Students need extended practice with constructions, since constructions embody the elements of geometry—lines and circles—independent of numbers and measurement. Since constructions are so central to Euclidean geometry, they are often called Euclidean constructions. Note: These constructions are basic in the sense that other important constructions introduced in later grades (e.g., of an equilateral triangle given one side; of a square inscribed in a circle) build on them. Use informal arguments such as paper folding to verify the correctness of constructions. G.5.3 Recognize and work with simple polyhedra. a. Represent and work with rectangular prisms by means of orthogonal views, projective views, and nets. A net is a flat (two-dimensional) pattern of faces that can be folded to form the surface of a solid. Note: Because a net represents the surface of a polyhedra spread out in two dimensions, the area of a net equals the surface area of the corresponding solid.

Orthogonal views are from top, front, and side; picture views are either projective or isometric; and nets are plane figures that can be folded to form the surface of the solid. Example: An orthogonal view (a), a projective view (b), and a net (c) of the same rectangular prism: b. Recognize the five regular ("Platonic") solids. Count faces, edges, and vertices, and make a table with the results. G.5.4 Find the area of shapes created out of triangles. a. Understand, derive, and use the formula A = ½bh for the area of a triangle. Arrange two identical right triangles with base b and height h to form a rectangle whose area is bh. Since the area of each right triangle is half that of the rectangle, A = ½bh. If triangle ABC is not a right triangle, then placing two copies together will form a parallelogram with base b and height h. This parallelogram can be transformed into a rectangle of area bh by moving a right triangle of height h from one side of the parallelogram to the other. So here too, A = ½bh.

Alternatively, divide a general triangle ABC into two right triangles as shown below and combine the areas of the two parts: Note: As the diagrams show, there are two cases to consider: For an acute triangle (where all angles are smaller than a right angle), the parts are added together. For an obtuse triangle (where one angle is larger than a right angle), one right triangle must be subtracted from the other). b. Find the area of a convex polygon by decomposing it into triangles. A polygon is called convex if a line segment joining any two points on the perimeter of the polygon will lie inside or on the polygon. Any convex polygon of n sides can be decomposed into (n – 2) triangles. c. Find the area of other geometric figures that can be paved by triangles. G.5.5 Interpret and plot points on the coordinate plane. a. Associate an ordered pair of numbers with a point in the first (upper right) quadrant and, conversely, any such point with an ordered pair of numbers. Positions on the coordinate plane are determined in relation to the coordinate axes, a pair of number lines that are placed perpendicular to each other so that the zero point of each coincides.

Note: The coordinate plane is a two-dimensional extension of the number line and builds on extensive (but separate) prior work with the number line and with perpendicular lines. Recognize the similarity between locating points on the coordinate plane and locating positions on a map. Recognize and use the terms vertical and horizontal. b. Identify characteristics of the set of points that define vertical and horizontal line segments. Use subtraction of whole numbers, fractions, and decimals to find the length of vertical or horizontal line segments identified by the ordered pairs of its endpoints. Example: What is the length of the line segment determined by (3/5, 0) and (1.5, 0)? Grade 6 G.6.1 Understand and use basic properties of triangles and quadrilaterals. a. Understand and use the angle properties of triangles and quadrilaterals. The sum of angles in a right triangle is 180°, since two identical right triangles form a rectangle. Note: By definition, a rectangle has 4 right angles, so the sum of the angles of a rectangle is 4 x 90° = 360°. Each right triangle contains half 360°, or 180°.

Note: Since one angle in a right triangle is 90°, the sum of the remaining two angles is also 90°. Since the sum of angles in a parallelogram is also 360°, the sum of angles in any triangle is also 360°/2 = 180°. Note: Following the line of argument used in grade 5 to find the area of a triangle, we note that (a) two identical copies of any triangle can be arranged to form a parallelogram, and (b) any parallelogram can be transformed into a rectangle with the same angle sum by moving a triangle from one side of the parallelogram to the other. Note: Alternatively, following the secondary argument offered in grade 5, one can drop a perpendicular to divide any triangle into two right triangles. The sum of the interior angles of each of these right triangles is 180°, but when put together they include two superfluous right angles. Subtracting these yields 180° + 180° – (90° + 90°) = 180° as the sum of the interior angles of any triangle.

Since any quadrilateral can be divided into two triangles the sum of the angles in a quadrilateral is also 2 × 180° = 360°. b. Use a protractor, ruler, square, and compass to draw triangles and quadrilaterals from data given in either numerical or geometric form. Draw a variety of triangles (right, isosceles, acute, obtuse) and quadrilaterals (squares, rectangles, parallelograms, trapezoids) of different dimensions. Verify basic properties of triangles and quadrilaterals by direct measurement. Note: Verification by measurement requires many examples, especially some with relatively extreme or uncommon dimensions. Examples: In parallelograms, opposite sides and opposite angles are equal; in rectangles, diagonals are equal. Example: Cut any triangle out of paper and tear it into three parts so that each part contains one of the triangle’s vertices. Notice that when the angles are placed together, the edge is straight (180°). Explore properties of triangles and quadrilaterals with dynamic geometry software. Note: Is a parallelogram a trapezoid? It depends on the definition of trapezoid.

If a trapezoid is defined as a quadrilateral with at least one pair of parallel edges, parallelograms become special cases of trapezoids. However, dictionaries usually define a trapezoid as a quadrilateral with exactly one pair of parallel edges, thereby distinguishing between parallelograms and trapezoids. Mathematicians generally prefer nested definitions as conditions become more or less restrictive. For example, all positive whole numbers are integers, all integers are rational numbers, and all rational numbers are real numbers. So in the world of mathematics, squares are rectangles, rectangles are parallelograms, and parallelograms are trapezoids. G.6.2 Understand and use the concepts of translation, rotation, reflection, and congruence in the plane. a. Recognize that every rigid motion of a polygon in the plane can be created by some combination of translation, rotation, and reflection. Translation, rotation, and reflection move a polygonal figure in the plane from one position to another without changing its length or angle measurements. Explore the meaning of rotation, translation, and reflection through drawings and hands-on experiments.

Example: The figure below illustrates how a rigid motion can be decomposed into a series of three steps: translation, reflection, and rotation. b. Understand several different characterizations and examples of congruence. Two figures in the plane are called congruent if they have the same size and same shape. Two shapes are congruent if they can be made to coincide when superimposed by means of a rigid motion. Two polygons are congruent if they have the same number of sides and if their corresponding sides and angles are equal. Note: Historically—beginning with Euclid—congruence applied only to polygons and used this as the definition. Indeed, the important properties of congruence are typically only about polygonal figures. Congruent figures in the plane are those that can be laid on top of one another by rotations, reflections, and translations. Note: Using rotations, reflections, and translations to define congruence gives precise meaning to the intuitive idea of congruence as "same size and same shape," thus permitting a precise definition of congruence for shapes other than polygons. Note: Technological aids (transparencies, dynamic geometry programs) help greatly in studying rigid motions. c.

Identify congruent polygonal figures. Understand why the two triangles formed by drawing a diagnonal of a parallelogram are congruent. Understand why the two triangles formed by bisecting the vertex angle of an isosceles triangle are congruent. G.6.3 Understand and use different kinds of symmetry in the plane. a. Symmetries in the plane are actions that leave figures unchanged. Explore and explain the symmetry of geometric figures from the standpoint of rotations, reflections and translations. b. Identify and utilize bilateral and rotational symmetry in regular polygons. A regular polygon is a polygon whose sides and angles are all equal. Bilateral symmetry means there is a reflection that leaves everything unchanged. Regular polygons have rotational symmetries. c. Identify and utilize translational symmetry in tessellations of the plane. Most tessellations have translation symmetry. d. Use reflections to study isosceles triangles and isosceles trapezoids. Understand why the base angles of an isosceles triangle are equal. Understand why the bisector of the angle opposite the base is the perpendicular bisector of the base. Note: Draw an angle bisector on an isosceles triangle.

Fold the drawing along the angle bisector (that is, reflect across the angle bisector). Then the base vertices collapse on each other: Both angles are equal, thus the angle bisector also bisects the base. Geometry Glossary Math Help > Geometry > Geometry Glossary Glossary of Geometrical Terms Have you ever imagined a geometrical shape, and then had trouble describing it? Many shapes have names, but they're fairly obscure, so a glossary might help. General notes: planes, lines, cones, and cylinders extend infinitely in both (or all) directions. A cone, in particular, is not shaped like an "ice cream cone" but rather like two such objects, attached at their pointy ends, and extending infinitely in both directions. Shapes that are bounded are called "line segments", "frustums", "solid cones", "prisms", etc. So read the definitions carefully.

(See also thesaurus and mathworld ) acute angle - an angle whose measure is greater than 0 but less than 90 degrees adjacent angles - 2 nonstraight and nonzero angles that have a common side in the interior of the angle formed by the noncommon sides alternate exterior angles - exterior angles on alternate sides of the transversal (not on the same parallel line) alternate interior angles - interior angles on alternate sides of the transversal (not on the same parallel line) angle - the union of 2 rays that have the same endpoint; measured in degrees or radians; the five types of angles are zero, acute, right, obtuse, and straight angle bisector - a ray that is in the interior of an angle and forms two equal angles with the sides of that angle annulus - a ring; the area bounded by two concentric circles apothem - in a regular polygon, the perpendicular distance from the center to a side; in a circle with a chord, the distance from the midpoint of a chord to the circle's center arbelos arbelos - a plane region bounded by three mutually tangent semicircles whose diameters are collinear.

arc - a continuous portion (as of a circle or ellipse) of a curved line arc length - the distance between an arc's endpoints along the path of the circle area - the surface enclosed within a closed plane figure; the measure of the surface, expressed in equivalent square units, such as square inches. axis - a straight line about which a body or a geometric figure rotates or may be supposed to rotate; a straight line with respect to which a body or figure is symmetrical -- called also axis of symmetry; one of the reference lines of a coordinate system ball - a "solid sphere"; the interior of a sphere (open ball); a sphere and its interior (closed ball); Borromean rings Borromean rings Borromean rings - A set of three rings joined in such a way that no pair is interlinked, but the three cannot be separated. bundle - the set of planes through a point, in projective geometry . cage - regular graph that has as few vertices as possible for its girth. Formally, an (r,g)-graph is defined to be a graph in which each vertex has exactly r neighbors, and in which the shortest cycle has length exactly g.

The (r,3)-cage is a complete graph Kr+1 on r+1 vertices, and the (r,4)-cage is a complete bipartite graph Kr,r on 2r vertices. Other cages listed in Wikipedia include the Moore graphs: (3,5)-cage: Petersen graph, (3,6)-cage: Heawood graph, (3,8)-cage: Tutte�Coxeter graph, (7,5)-cage: Hoffman�Singleton graph. Cartesian plane - a coordinate plane central angle of a circle - an angle whose vertex is the center of the circle cevian - a line segment connecting a vetex of a triangle with a point on the opposite side of the triangle. The condition for three general Cevians from the three vertices of a triangle to concur (at a cevian point) is known as Ceva's theorem . cevian point - the point of concurrence of three cevians cevian triangle - Given a point P and a triangle ABC, the Cevian triangle A'B'C' is defined as the triangle composed of the endpoints of the cevians through the Cevian point P.

chord - In general, a straight line joining two points on a curve; often, chord is used to mean a straight line segment joining and included between two points on a circle; chord of a circle - a segment whose endpoints are on a circle circle - a closed plane curve every point of which is equidistant from a fixed point, called the center (the center is not part of the circle) circular - having the shape of a circle; a circular cylinder is one in which its defining shape is a circle, and most often is used to mean a right circular cylinder; a circular cone is one in which its defining shape is a circle, and most often is used to mean a right circular cone; complementary angles - 2 angles whose measures, when added together, equal 90 degrees complex polygon - a polygon having at least one pair of intersecting edges; as opposed to simple; reflex polygon. concave set - a set of points in which not all segments connecting points of the set lie entirely in the set; nonconvex. concentric - having the same center, as concentric circles; having the same axis, as concentric cylinders cone - in general, a cone is the locus of (i.e.

surface traced by) the surface formed by lines joining every point of the boundary of a fixed planar closed curve (the base) to a common vertex; commonly, a right circular cone; a "solid cone" is a solid (or the space) bounded by the planar closed curve, called the base, and the line segments connecting the base to the vertex. The area of a solid cone is (1/3)Ah, where A is the area of the base, and h is the height. congruent - equal; exactly the same (size, shape, etc.). Two figures are congruent when one is the image of the other under a reflection or composite of reflections conic section - the figure formed by the intersection of a plane and a (right circular) cone consecutive sides - sides of a polygon that share an endpoint consecutive vertices - endpoints of a single side of a polygon constant width - of a bounded figure: having the same width in every direction. A Reuleaux triangle is an example of a figure of constant width. Other examples include the shape formed by an equilateral triangle of side s with circles of radius r around each vertex, and arcs of radius s+r centered on the "far" vertices that join these circles.

The same construction can be made with an equilateral (but not necessarily equiangular n-sided star, with n an odd number. See Cut-the-knot Shapes of Constant Width and Cut-the-knot Barbier's Theorem for more information. convex set - a set of points in which all segments connecting points of the set lie entirely in the set; There are three things one can do to see if a figure is convex - look for "dents", extend the segments (they shouldn't enter the figure), and connect any two points within the figure with a segment (if any part of the segment lies outside the figure, it's concave) corresponding angles - any pair of angles in similar locations with respect to a transversal coterminal angles - two angles that have the same terminal side; two angles that differ by an integer multiple of 360 degrees. cyclic quadrilateral - a quadrilateral whose four vertices lie on a circle. cyclogon cyclogon - a curve traced by any vertex of a regular polygon that rolls without sliding on a straight line.

The area under one arch of the cyclogon (pink, in this diagram) is equal to the area of the polygon plus twice the area of the circle that circumscribes the polygon. If the "vertices" of the cyclogon were linked by straight lines instead of curved ones, then the area under one arch of that figure would be three times the area of the regular polygon. cycloid cycloid - a curve traced by any point on the circumference of a circle that rolls without sliding on a straight line. The area under one arch of the cycloid (blue, in the diagram) is exactly three times the area of the circle. A curtate cycloid is a curve generated in the same way, except that the point is inside the circle rather than on the circle. A prolate cycloid is a curve generated in the same way, except that the point is rigidly fixed outside the circle. All cycloids, including curtate and prolate cycloids, are trochoids. cylinder - in general, a cylinder is the locus of (i.e.

surface traced by) a straight line moving parallel to a fixed straight line and intersecting a fixed planar closed curve (the base); commonly, a right circular cylinder diagonal - a line segment whose endpoints are 2 nonconsecutive vertices of a polygon. diameter of a circle (or sphere) - the line segment whose endpoints are points on a circle (or sphere) that contains the center of the circle as its midpoint; the length of that line segment disk - the interior of a circle (an open disk); a circle and its interior (a closed disk) ellipsoid - a squashed or stretched sphere in which each of the three axes can be of different lengths. (Contrast to a spheroid, in which two of the three axes have the same length.) An ellipsoid has the equation xÂ²/aÂ² + yÂ²/bÂ² + zÂ²/cÂ² = 1 epicycloid epicycloid - a curve generated by the motion of a point on the circumference of a circle that rolls externally, without slipping, on a fixed circle.

Equation: x = (a + b) cos(θ) − b cos[(a + b)θ/b] and y = (a + b) sin(θ) − b sin[(a + b)θ/b]. . . . . . . diagram needs rework. epitrochoid - a curve generated by the motion of a point rigidly connected with -- but not on the circumference of -- a circle that rolls externally, without slipping, on a fixed circle. equianglular - having angles of the same measure equidistant - the same distance from something equilateral - having sides that are equal in length even node - a node that has an even number of arcs exterior angle of a polygon - angle formed by one side and the extension of an adjacent side. exterior angles - angles outside of two lines cut by a transversal. face - a polygonal region of a surface frustum - the basal part of a solid cone or pyramid formed by cutting off the top by a plane parallel to the base; the part of a solid intersected between two usually parallel planes graph - abstract representation of a set of objects (nodes) where some pairs of the objects are connected by links (edges).

See also cage, snark, and lattice. Graphs listed in Wikipedia include Bull, Chv�tal, Clebsch, Desargues,D�rer, Folkman, Foster, Franklin, Frucht, Gray, Gr�tzsch, Hall�Janko, Harries, Harries-Wong, Heawood, Higman�Sims, Hoffman�Singleton, McGee, M�bius�Kantor, Null, Pappus, Petersen, Rado, Shrikhande, Tietze's, Tutte�Coxeter; Cages: Balaban 10-cage, Balaban 11-cage; Snarks: Flower, Szekeres; Lattices: Young�Fibonacci great circle - the circle formed by the intersection of a sphere and a plane that contains its center and that divides the sphere into two hemispheres hemisphere - half of a sphere hypocycloid - a curve generated by the motion of a point on the circumference of a circle that rolls internally (on the concave side), without slipping, on a fixed circle.

hypotenuse - the side opposite the right angle in a right triangle hypotrochoid - a curve generated by the motion of a point rigidly connected with -- but not on the circumference of -- a circle that rolls internally (on the concave side), without slipping, on a fixed circle. icosahedron - a 20-sided solid; each side is in the shape of a triangle included angle - the interior angle made by two consecutive sides of a polygon included side - the side between two consecutive angles in a polygon initial side - the side that the measurement of an angle starts from interior angles - angles between two lines cut by a transversal isosceles trapezoid - a trapezoid that has a pair of equiangular base angles. ( classification of quadrilaterals ) isosceles triangle - a triangle having two sides of equal length kite - a quadrilateral that has two distinct pairs of consecutive equilateral sides Klein Bottle klein bottle - the 3-dimensional analog of the Mobius Strip. The figure is impossible in three dimensions, because one part of its surface passes through another part of its surface without touching it. More on Wikipedia and Mathworld .

lattice - a regular geometrical arrangement of points over an area or in space lattice graph - a graph whose nodes form a lattice lattice points - points in the coordinate plane having integer coordinates leg of a right triangle - a side of a right triangle that include the 90 degree angle lemniscate - a figure that looks like the "infinity" symbol; a lazy 8; an 8 on its side. This is a cross-section of a torus (a toric section) taken on a plane tangent to the inner circle of the torus. In polar coordinates, the equation of a lemniscate is r=sqrt(cos(2θ)). See mathworld . linear pair of angles - two supplementary adjacent angles whose noncommon sides form a line locus - the set of points that satisfy a given condition or equation major arc - an arc whose endpoints form an angle over 180 degrees with the center of the circle; as opposed to minor arc; major axis (ellipse) - the line segment containing the foci of the ellipse and whose endpoints are on the ellipse. measure of an angle - the size of an angle, typically measured in degrees measure of an arc - The measure of the central angle subtended by an arc.

median - the segment connecting the vertex of an angle in a triangle to the midpoint of the side opposite it midpoint - the point M on line segment AB such that AM = AB. minor arc - an arc whose endpoints form an angle less than 180 degrees with the center of the circle. minor axis (ellipse) - the perpendicular bisector of the line segment whose endpoints are the foci of the ellipse. M�bius Strip M�bius Strip (sometimes spelled Moebius or Mobius) - a surface having only one side and only one edge. It can be formed from a rectangle by giving it a half twist, then gluing the edges together. mouhefanggai - the solid formed as intersection of two perpendicular right circular cylinders of equal diameter. This is also known as a Steinmetz Solid or a bicylinder. Cross sections of a mouhefanggai are square if they are taken parallel to the plane of the axes of the two intersecting cylinders. mouhefanggoid mouhefanggoid - the solid formed as the intersection of two perpendicular cylinders of elliptical cross-section and equal major or minor axes aligned.

The multi-colored diagram to the right illustrates half a mouhefanggoid formed by the intersection of a circular cylinder whose axis is parallel to the y axis with an elliptical cylinder whose axis is parallel to the x axis. The major axis of the ellipse is twice the minor axis, so cross-sections of this half- mouhefanggoid taken parallel to the x-y plane are squares. n-gon - a polygon having n sides. nonagon - a nine-sided polygon. nonconvex set - a set of points in which not all segments connecting points of the set lie entirely in the set; concave non-Euclidean geometry - geometry in which given a line and a point on the line, there is either no line (elliptic geometry) or infinitely many lines (hyperbolic geometry) parallel to the given line that pass through the given point. non-included side - the side of a triangle that is not included by 2 given angles. normal - perpendicular. In particular, a line is normal to a plane oblique prism or cylinder - a non-right prism or cylinder. obtuse angle - an angle whose measure is greater than 90 but less than 180 degrees.

octagon - an eight-sided polygon opposite rays - two rays having a common endpoint that form a line. parallelepiped - a prism whose opposite faces are all parallelograms and congruent (in pairs) parallelogram - a quadrilateral such that both pairs of opposite sides parallel pencil - the set of lines through a point, in projective geometry . pentagon - a five-sided polygon simple polygon - a polygon whose edges don't cross one another; as opposed to complex; skew lines - non-coplanar lines. Lines that are neither parallel nor intersecting. slant height - the length of a lateral edge of a cone snark - is a connected, bridgeless cubic graph with chromatic index equal to 4. In other words, it is a graph in which every vertex has three neighbors, and the edges cannot be colored by three colors without two edges of the same color meeting at a point. Snarks listed in Wikipedia include Petersen graph, Flower snark, Szekeres snark, Tietze's graph, Blanu�a snarks, Descartes' snark, double star snark.

solid - the union of the surface and the region of space enclosed by a 3-D figure; examples: conic solid, cylindric solid, rectangular solid solid geometry - the study of figures in three-dimensional space. sphere - a closed surface in three-dimensional space, every point of which is equidistant from a fixed point, called the center (the center is not part of the sphere) sphericon - a "cone with a twist". It is created by starting with two identical right circular solid cones (of a particular shape) joined at the base, then slicing them along a plane through both vertices, and then rotating one of the two pieces a quarter turn, and gluing them together. spheroid - an ellipsoid in which two of the three axes are equal. (Contrast to an ellipsoid, in which all three axes may have different lengths.) A spheroid has the equation (xÂ²+yÂ²)/aÂ² + zÂ²/cÂ² = 1. An oblate spheroid has a polar axis that is shorter than the diameter of the equatorial circle, and can be formed by rotating an ellipse about its minor axis.

A prolate spheroid has a polar axis that is longer than the diameter of the equatorial circle, and can be formed by rotating an ellipse about its major axis. square - an equilateral and equiangular quadrilateral straight angle - an angle whose measure is 180 degrees; an angle whose sides are collinear. straight line - a collection of points of such a nature that if one picks any three of them, A, B, and C, the distances AB, AC, and BC will be such that the sum of two of them will equal the third. supplementary angles - two angles whose measures sum to 180 degrees. surface - the boundary of a 3-D figure. symmetry diagonal - the diagonal of a kite (a kind of quadrilateral ) that perpendicularly bisects the other. symmetry line - the line of reflection in a reflection-symmetric figure. terminal side - the side that the measurement of an angle ends at. tesselation - a covering of a plane with congruent copies of the same region with no holes or overlaps. tesseract - the four-dimensional analog of the cube; a 4-D hypercube.

tetragon - a four-sided polygon; another (rather unusual) word for quadrilateral three dimensional figure - a set of points in space; examples: box, cone, cylinder, parallelpiped, prism, pyramid, regular pyramid, right cone, right cylinder, right prism, sphere. torus - a surface or solid shaped like a doughnut and formed by revolving a circle about a line in its plane that does not intersect it. transversal - a line that intersects two other (usually parallel) lines. trapezoid - a quadrilateral that has at least one pair of parallel sides triangle - a polygon having three sides. trochoid - a curve traced by a point on a radius or an extension of the radius of a circle that rolls, without slipping, on a curve, another circle, or a straight line. Equation: x = aθ − b sin θ, y = a − b cos θ; any of a class of curves including the cycloid, epicycloid, epitrochoid, hypocycloid, and hypotrochoid. Cf: roulette, in which the rolling circle is replaced by any curve.

vertex angle - the angle formed by the equilateral sides of an isosceles triangle. vertex of a polygon - an endpoint of a segment in a polygon. vertex of an angle - the common endpoint of the two rays. vertical angles - 2 angles that share a common vertex and whose sides form 2 lines. width - of a bounded shape: the distance between two parallel lines, each touching the boundary of the shape but not its interior. This is called "the width of the shape in the direction of the lines". Note that figures of constant width (such as a Reuleaux triangle) have the same width in every direction. zero angle - an angle whose measure is 0. In a zero angle, both the initial and terminal sides are the same. Internet references Cut-the-knot Shapes of Constant Width has an applet showing how the Reuleaux triangle has constant width. A "Reuleaux drill" can't make a square hole, but it can make a hole that's almost square, carving out almost 99% of the area of the square. Cut-the-knot Barbier's Theorem for more applets and a general method of making shapes of constant width.

Barbier's Theorem states that the circumference of any shape of constant width is equal to πD, where D is the width of the shape.

Question: Which solid can be generated by rotating an isosceles triangle about its line of symmetry?
Answer: CONE