# The radius for a regular polygon is the same as the radius of the circumscribed circle.

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1 Perimeter and Area The perimeter and area of geometric shapes are basic properties that we need to know. The more complex a shape is, the more complex the process can be in finding its perimeter and area. Perimeter: The distance around a shape. Circumference: The distance around a circle. Area: The amount of surface covered by a figure. Center (of the polygon): The center of the circumscribed circle. Radius (of the polygon): The radius of the circumscribed circle. Apothem: A perpendicular segment from the center to a side of the polygon. Units Perimeter The perimeter is the sum of all the edges of a two-dimensional figure. The perimeter is measured in units of length (e.g. feet, inches, centimeter). If the unit is not specified, the perimeter is measured in units. Circumference The circumference is also measured in units of length (e.g. feet, inches, centimeter). If the unit is not specified, the circumference is measured in units. Area The area is measured in square units (e.g. square feet or ft 2 ). If the unit is not specified, the area is measured in units 2. Regular Polygons A regular polygon is a polygon that is equiangular (all angles are equal) and equilateral (all sides have the same side lengths). All regular polygons can be inscribed in a circle, so these polygons also have a center and a radius. The radius for a regular polygon is the same as the radius of the circumscribed circle. When the regular polygon is inscribed in a unit circle, the radius is 1. A central angle is the angle formed by two radii drawn to consecutive vertices of the polygon. The angle measure is Length of an apothem = r is the length of the radius n is the number of sides Perimeter of a Regular Polygon The simplest way to find the perimeter of a regular polygon would be to just add the lengths of all the sides. P = ns n = number of sides s = side lengths There is another version of the perimeter formula: n = number of sides r = radius length Page 1 of 2 v

2 Perimeter and Area cont. Regular Polygons (cont.) Area of a Regular Polygon Theorem: Area = P = perimeter a = apothem The area of the triangle is triangle. Can also be written as: Area = n = number of sides r = radius length Regular Properties Polygons of Area (cont.) and Perimeter as, so the area of the regular polygon, which has n triangles, is n times the area of the Congruent Areas Postulate: If two figures are congruent, they have the same area. The converse is not true! Two figures with the same area do not have to be congruent. If the polygon is not a regular polygon, the area can be found by dividing the polygon into smaller polygons where the areas can be calculated. Area Addition Postulate: If a figure is composed of two or more parts that do not overlap each other, then the area of the figure is the sum of the areas of the parts. If you remember the formulas for perimeter and area for rectangles and triangles, you can always divide other shapes into rectangles and triangles. Perimeter of Similar Polygons Theorem: If two polygons are similar, then the ratio of the perimeters is equal to the ratio of the corresponding side lengths. If ABCD ~ QRST, then. Area of Similar Polygons Theorem: If the scale factor of the sides of two similar polygons is, then the ratio of the areas would be. If ABCD ~ QRST and is the scale factor, then the ratio of the areas is. Page 2 of 2

3 Triangles and Quadrilaterals Triangles and quadrilaterals are among the more basic and common polygons. Triangles always have interior angles sum to 180 while quadrilaterals always have interior angles sum to 360. Perimeter: The distance around a shape. Area: The amount of surface covered by a figure. Square Perimeter = Postulate: The area of a square is the square of the length of its side. Area = Rectangle Perimeter = Theorem: The area of a rectangle is the product of its base and height. Area = Parallelogram Either pair of parallel sides can be the bases of a parallelogram. The height is perpendicular to the base - the side is NOT the height! Area = The area of a parallelogram is the same as the area of a rectangle. Page 1 of 2 v

4 Triangles and Quadrilaterals cont. Triangle Perimeter = Theorem: The area of a triangle is one half the product of the base and its corresponding height. Area = If a parallelogram is cut in half along a diagonal, there would be two congruent triangles. The area of the triangle, then, is half the area of the area of a parallelogram. Trapezoid The height of a trapezoid is the perpendicular distance between its bases. Theorem: The area of a trapezoid is one half the product of the height and the sum of the lengths of the bases. Area = A trapezoid can be turned into a parallelogram with height h and base. The area for the parallelogram is. So the area of the trapezoid is half the parallelogram: Rhombus and Kite Both rhombuses (left) and kites (right) have perpendicular diagonals. Rhombus Theorem: The area of a rhombus is half the product of the lengths of the diagonals. Kite Theorem: The area of a kite is half the product of the lengths of the diagonals. The formulas for the areas of rhombus and kite are the same! The areas for rhombus and kite can be found by creating two rectangles: The area of the rectangles is:. Page 2 of 2

5 Circumference and Arc Length The circumference and arc of a a circle can be found with the appropriate theorems and formulas. Circumference: The distance around a circle. Sector (of a circle): The area bounded by two radii and the arc between the endpoints of the radii. Segment (of a circle): The area of a circle that is bounded by a chord and the arc with the same endpoints as the chord. Parts of a Circle Circle: Set of all points in a plane that are a given distance from another point (the center). Circumference: Perimeter of a circle. Radius: Any segment from the center to a point on the circle (written as r). Diameter: Any segment from one point on the circle through the center to another point on the circle (written as d). d = 2r Circumference The circumference of a circle is just the perimeter of a circle. Circumeference = 2πr = πd π (an irrational number pronounced pi ) Area Theorem: The area of a circle is π times the square of the radius. Area = πr 2 Arc Length Arcs are fractional portions of circles. They are measured in degree measures and linear measures. 1. Degree measure: fractional part of a 360 complete circle that the arc is in 2. Linear measure: length of the arc Corollary: The ratio of the arc length to the circumference is equal to the ratio of the arc measure to 360. length of AB Page 1 of 2 v

6 Circumference and Arc Length cont. Sectors The sector is a fractional part of the area of the circle, often written as fraction of the circle. Finding the area of the sector is like finding the fractional part of the area of the circle. Theorem: Area of a sector = Basically, the fractional portion of the circle that the sector circle, giving the area of the circle. If the arc length (s) is known,. represents is multiplied by the total area of the Segments The area of the segment is the area of the sector minus the area of the triangle made by the radii. Do not confuse the segment of a circle with line segment! Notes Page 2 of 2

7 Polyhedra The 2-dimensional shapes of a polygon can be applied in a 3-dimensional figure. Such characteristics define polyhedra. Polyhedron is a very general terms and can include some very complex shapes. Polyhedron (plural, polyhedra): A three-dimensional figure made up with polygon faces. Face: A polygon in a polyhedron. Lateral Face: A face that is not the base. Edge: The line segment where two faces intersect. Lateral Edge: The line segment where two lateral faces intersect. Vertex (plural, vertices): The point where two edges intersect. Regular Polyhedron: A polyhedron where all the faces are congruent regular polygons. Classifying Polyhedra A polyhedron has these properties: 1. 3-dimensional 2. Made of only flat polygons, called the faces of the polyhedron 3. Polygon faces join together along segments called edges 4. Each edge joins exactly two faces 5. Edges meet in points called vertices; each edge joins exactly two vertices 6. There are no gaps between edges or vertices 7. Can be convex or concave Two common types of polyhedra include prisms and pyramids. Prisms and pyramids are named by their bases. Prism: A polyhedron with two parallel, congruent bases. The other faces, also called lateral faces, are formed by connecting the corresponding vertices of the bases. Left: triangular prism Right: octagonal prism Pyramid: A polyhedron with one base and triangular sides meeting at a common vertex. Left: hexagonal pyramid Right: square pyramid Page 1 of 2 v

8 Polyhedra cont. Euler s Formula for Polyhedra This formula can be used to find the number of vertices (V), faces (F), or edges (E) on a polyhedron: F + V = E + 2 If a figure does not satisfy Euler s formula, the figure is not a polyhedron. Regular Polyhedra A regular polyhedron has the following characteristics: 1. All faces are congruent regular polygons 2. Satisfies Euler s formula for the number of vertices, faces, and edges 3. The figure has no gaps or holes 4. The figure is convex (has no indentations) Platonic Solids Named after the Greek philosopher Plato, the five regular polyhedra are: 1. regular tetrahedron: 4-faced polyhedron where all the faces are equilateral triangles 2. cube: 6-faced polyhedron where all the faces are squares 3. regular octahedron: 8-faced polyhedron where all the faces are equilateral triangles 4. regular dodecahedron: 12-faced polyhedron where all the faces are regular pentagons 5. regular icosahedron: 20-faced polyhedron where all the faces are equilateral triangles Semi-Regular Polyhedra A polyhedron is semi-regular if all of its faces are regular polygons and satisfies Euler s formula. Semi-regular polyhedra often have two different kinds of faces, both of which are regular polygons. Prisms with a regular polygon base are one example of semi-regular polyhedron. Notes Page 2 of 2

9 Representing Solids There are four main ways to visualize a three-dimensional figure in two dimensions: isometric view, orthographic view, cross-sectional view, and a net. Perspective: Artistic illusion used to make things in the distance look smaller by using a vanishing point where parallel lines converge. Isometric View: Three-dimensional view of a solid that does not typically include perspective. Orthographic Projection: A view that shows a flat representation of each side of the figure s sides. Cross Section View: A slice of a three-dimensional figure. Net: A two-dimensional figure that can be folded into a geometric solid. Isometric View The perspective view looks more real to the eye, but isometric view is more useful for measuring and comparing distances. It is often shown in a transparent form; shading and coloring can also be applied to make the figure look more realistic. solid see through dotted shaded Orthographic View How to show a figure in an orthographic projection: 1. Place it in an imaginary box. 2. Project each side of the figure out to each of the walls of the box. 3. The image of the side will be on each of the six walls of the box. For example: Page 1 of 2 v

10 Representing Solids cont. Cross Section View This is similar to slicing a 3-dimensional figure into a series of thin slices. Each slice will show a cross section view. Depending on the angle at which we slice the figure, there are many possible cross sections that we can get. Net Nets are just another way to model a figure. If a net is cut out, it can be folded into a model of a figure. A single figure can have multiple possible nets. Notes Page 2 of 2

11 Surface Area and Volume Surface area and volume are two very fundamental properties of 3-dimensional shapes. Often times in geometry we will be asked to find the surface area or volume of a shape. The more simple shapes can be solved by using a general formula. More complex shapes will require us to apply our knowledge of one or several 2-dimensional shapes. Surface Area: The sum of the areas of the faces. Lateral Area: The sum of the areas of the lateral faces only. Lateral Face: A face that is not the base. Volume: The measure of how much space a 3-dimensional figure occupies. Surface Area and Volume Surface Area Surface area can be calculated by constructing a net of the 3-dimensional figure and using the Area Addition Postulate. Area Addition Postulate: The surface area of a 3-dimensional figure is the sum of the areas of all its non-overlapping parts. The lateral area is the surface area of the 3-dimensional figure minus the area of the base(s). Volume Volume is measured in the cubic units. Two postulates that help us find the volume are: Volume Congruence Postulate: If two polyhedrons are congruent, then their volumes are congruent. Volume Addition Postulate: The volume of a solid is the sum of the volume of all of its non-overlapping parts. Volume can be found by counting boxes, which is a method where we count how many units, like building blocks, create our figure. The volume of the counting boxes can be found using the Volume of a Cube Postulate: Volume of a Cube Postulate: The volume of a cube is the cube of the length of its side, or s 3. For oblique figures where the lateral edges are not perpendicular to the base, the volume can be found by applying Cavalieri s Principle. Cavalieri s Principle: If two solids have the same height and the same cross-sectional area at every level, then the two solids have the same volume. Think of Cavalieri s Principle as a stack of books. Each book in the stack still has the same volume regardless of whether the stack is leaning or not. Page 1 of 2 v

12 Surface Area and Volume cont. Composite Solids Sometimes we will be asked to find the volume and surface area of a hollowed out figure. An example of this would be a pipe. Volume of a Composite Solid 1. Find the volume of the entire figure as if it had no hole. 2. Find the volume of the hole. 3. Subtract the volume of the hole from the volume of the entire cylinder. Surface Area of a Composite Solid 1. Find the lateral area of the larger figure. 2. Find the lateral area of the hole. 3. Find the area of the bases of the larger figure. 4. Find the area of the bases of the hole. 5. Subtract the area of hole s bases from the larger figure s bases. 6. Add to the lateral area of the larger figure and the lateral area of the hole. Similar Solids Similar solids are two solids of the same type with equal ratios of corresponding linear measures (for example, heights and radii). All ratios for corresponding measures must be the same. Characteristics of Similar Figures Recall that when two polygons are similar, the ratio relating any two corresponding lengths is equal to the scale factor. The ratio of the areas of two similar figures is then equal to the square of the ratio between the corresponding linear sides. For example, if we doubled the sides of a cube, the surface area would quadruple. The ratio of the volumes of two similar figures is equal to the cube of the ratio between the corresponding linear sides. Scale Factor for Similar Figures Ratios Units Scale Factor in, ft, cm, m, etc. Ratio of the Surface Areas in 2, ft 2, cm 2, m 2, etc. Ratio of the Volumes in 3, ft 3, cm 3, m 3, etc. Notes Page 2 of 2

13 Prisms and Cylinders Prisms and cylinders are among the simplest 3-dimensional objects. They have two parallel bases of equal size. We can find the volume and surface area of these objects by using formulas. Prism: A polyhedron with two parallel, congruent bases. Cylinder: A solid with congruent circular bases that are in parallel planes with the space between the circles is enclosed. Prisms A prism has a pair of parallel bases and rectangular lateral (non-base) faces. Prisms are named by their bases. If the lateral faces are perpendicular to the bases, the prism is a right prism. If the faces lean to one side, the prism is oblique. Surface of a Prism Theorem: The surface area of a prism is the sum of the area of the bases and the area of each lateral face. surface area = SA = lateral area + 2 area of base The lateral area is the sum of the area of all the lateral faces. Another way to calculate the lateral area for a right prism is: lateral area = perimeter height Another way to write the surface area for a right prism is: SA = perimeter height + 2 area of base Volume of a Prism Theorem: The volume of a prism is V = Bh, where B is the area of the base and h is the prism s height. For a rectangular prism, the base is a rectangle, so the volume formula can be rewritten as: V = lwh. For oblique prisms, Cavalieri s Principle holds, so the volumes of oblique prisms and right prisms have the same formula. The height for oblique prisms is the altitude outside the prism. Cylinders A cylinder is a 3-dimensional figure with a pair of parallel and congruent circular ends, or bases. Page 1 of 2 v

14 Prisms and Cylinders cont. Cylinders (cont.) Surface Area of a Right Cylinder We can deconstruct a cylinder into a net. The sum of the areas of all the components, the two bases and the lateral side, will give us the total surface area of the cylinder. The net for a right cylinder is a rectangle and two circular bases. The surface area can be found with the following equation: SA = (area of two bases) + (area of lateral side) This can also be written as: SA = 2 (πr 2 )+2πr h Theorem: The surface area of a right cylinder with radius r and height h is. Volume of a Cylinder Theorem: The volume of a cylinder is V = Bh, where B is the area of the base and h is the prism s height. The base of a cylinder is a circle, so the volume of a cylinder can be written as:. For oblique cylinders, Cavalieri s Principle holds, so the volumes of oblique cylinders and right cylinders have the same formula. The height for oblique cylinders is the altitude outside the prism. Notes Page 2 of 2

15 Pyramids and Cones Pyramids and cones have sides that join at a single point. The properties of pyramids and cones are often used in real life. With surprisingly simple formulas, we can find the volume and surface area of these objects. Pyramid: A polyhedron with one base and triangular sides meeting at a common vertex. Slant Height: The height of a lateral face of a regular pyramid. Cone: A solid with a circular base and sides tapering up towards a common vertex. Pyramids A pyramid has one base and all the lateral faces meeting at a vertex. When the vertex is directly above the center of the base and the base is a regular polygon, the pyramid is regular. A regular pyramid has a slant height, which is the height of the lateral face. Non-regular pyramids do not have a slant height. Surface Area of a Regular Pyramid Theorem: For a regular pyramid with a base of area B, perimeter P, and slant height l, then the surface area is: Volume of a Pyramid Theorem: The volume of a pyramid is, where B is the area of the base and h is the height. The volume of a pyramid is one-third the volume of a prism with the same base. Cones A cone is a solid with a circular base and sides that taper up towards a common vertex. The radius of the base is also the radius of the cone. If the segment connecting the vertex of the cone to the center of the base is perpendicular to the base, then the cone is a right cone. It has a slant height just like a pyramid; the slant height is the distance between the vertex and a point on the base. Surface Area of a Right Cone The surface area of a right cone can be tricky to calculate. The net of a cone is shown on the left. The area of the lateral face is a sector and can be found by using the following proportion: Theorem: The surface area of a right cone with base radius r and slant height l is. Volume of a Cone Theorem: The volume of a cone is, where r is the radius of a cone and h is the height. The volume of a cone is one-third the volume of a cylinder with the same base. Page 1 of 1 v

16 Spheres Spheres can be thought of as 3-dimensional circles. While all the points of a circle are equally far from the center in a 2-dimensional space, the equidistant points of spheres are in 3-dimensional space. Sphere: The set of all points in three-dimensional space that are equidistant (equally far) from a point. Characteristics of a Shpere A sphere is a 3-dimensional circle. Think of it like a ball. A sphere is the set of all points that lie a fixed distance r from a center point. A sphere is the surface that results when a circle is rotated about any of its diameters. A sphere results when you construct a polyhedron with an infinite number of faces that are infinitely small. Parts of a Sphere Center: the point at the center of a sphere; all points on the sphere lie equidistant from the center Diameter: length of segment connecting any two points on the sphere s surface and passing through the center Radius: the length of a segment connecting the center of the sphere with any point on the sphere s surface The radius is 1 2 the diameter Secant: line, ray, or line segment that intersects a circle or sphere in two places and extends outside of the circle or sphere Tangent: intersects the circle or sphere at only one point All tangents are perpendicular to the radii that intersect with them Great circle: plane that contains the diameter The great circle is the largest circle cross section There are infinitely many great circles Hemisphere: half a sphere A great circle divides a sphere into two congruent hemispheres Page 1 of 2 v

17 Spheres cont. Surface Area and Volume of a Sphere Theorem: The surface area of a sphere is SA = 4πr 2, where r is the radius. To remember the formula for the surface area of a sphere, think of a baseball. The cover of the baseball can be approximated by four circles. Each circle has an area of πr 2, so the surface area of the baseball is 4πr 2. r Theorem: The volume of a sphere is, where r is the radius. We can approximate the volume of the sphere by adding up the volumes of an infinite number of infinitely small pyramids, as illustrated below. Each of the faces is the base of a pyramid with the vertex located at the center. Since the volume of the pyramid is, the volume of the sphere is: The sum of all the bases is the surface area of the sphere. The formula for the volume of a sphere can then be simplified as: Notes Page 2 of 2

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