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Areas and Volumes of Solids Chapter 12 Section 12-1 Prisms POLYHEDRON is a three-dimensional figure in which each surface is a polygon and The surfaces are called faces. Two faces intersect at an edge, and a vertex is a point where three or more edges intersect. PRISM Is a polyhedron with two identical parallel faces. Each of these faces is called a base and a prism is named by the shape of its bases. ALTITUDE • A segment joining the two base planes and perpendicular to both • The length of the altitude is the height of the prism LATERAL FACES faces that are not bases RIGHT PRISM A prism having rectangular lateral faces. OBLIQUE PRISM A prism having nonrectangular lateral faces. THEOREM 12-1 The lateral area of a right prism equals the perimeter of a base times the height of the prism LA = ph SURFACE AREA OF A PRISM Surface Area equals the sum of the areas of all its faces SA = LA +2B * Also called Total Area THEOREM 12-2 The volume of a right prism equals the area of a base times the height of the prism. V = Bh Section 12-2 Pyramids PYRAMID Is a polyhedron with only one base. The other faces are triangles that meet at a vertex and a pyramid is named by the shape of its base. ALTITUDE Is a segment from the vertex perpendicular to the base. LATERAL FACES are the triangular faces. LATERAL EDGES The segments where the lateral faces intersect. REGULAR PYRAMID Base is a regular polygon All lateral edges are congruent All lateral faces are congruent isosceles triangles The height is called the slant height The altitude meets the base at its center METHODS FOR FINDING LATERAL AREA OF A REGULAR PYRAMID 1. Find the area of one lateral face and multiply by n where n is the number of lateral faces 2. Use the formula LA = ½pl where l is the slant height THEOREM 12-4 The volume of a pyramid equals 1/3 the area of the base times the height of pyramid. V = 1/3Bh Section 12-3 Cylinders and Cones CYLINDER a three-dimensional figure having a curved region with two parallel congruent circular bases. Its axis joins the centers of the two bases. RIGHT CYLINDER Is a cylinder where the segment joining the center of the circular bases is an altitude ALTITUDE Is the height of the cylinder THEOREM 12-5 The lateral area of a cylinder equals the circumference of a base times the height of the cylinder. LA = 2rh THEOREM 12-6 The volume of a cylinder equals the area of a base times the height of the cylinder. 2 V = r h CONE a three-dimensional figure having a curved surface and one circular base. Its axis is a segment from the vertex to the center of the base. THEOREM 12-7 The lateral area of a cone equals half the circumference of the base times the slant height LA = rl THEOREM 12-8 The volume of a cone equals one third the area of the base times the height of the cone. 2 V = 1/3r h Section 12-4 Spheres SPHERE is the set of points in space that are the same distance from a given point called the center THEOREM 12-9 The area of a sphere equals 4 times the square of the radius 2 A = 4r THEOREM 12-10 The volume of a sphere equals 4/3 times the cube of the radius 3 V = 4/3r Section 12-5 Areas and Volumes of Similar Solids SIMILAR SOLIDS Solids having the same shape but not necessarily the same size THEOREM 12-11 If the scale factor of two similar solids is a:b, then 1. The ratio of corresponding perimeters is a:b 2. The ratio of the base areas, of the lateral areas, and of 2 2 the total areas is a :b 3. The ratio of the volume is a3:b3 END

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