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Math 53 "Winter ’09" 8.1 "Prisms, Area, and Volume" ————————————————————————————————————————————————— Objectives: * Finding the area and volume of prisms ————————————————————————————————————————————————— Prisms If two congruent polygons lie in parallel planes in such a way that their corresponding sides are parallel and if the corresponding vertices of these polygons are joined by line segments, then the "solid" that results is a prism. De…nition: "Right Prism" A right prism is a prism in which the lateral edges are perpendicular to the base edges at their points of intersection. De…nition: "Oblique Prism" An oblique prism is a prism in which the parallel lateral edges are oblique to the base edges at their points of intersection. Area of a Prism De…nition: "Lateral Area" kThe lateral area L of a prism is the sum of the areas of all lateral facesk Theorem 8.1.1: The lateral area L of any prism whose altitude has measure h and whose base has perimeter P is given by: Page: 1 Bibiana Lopez Elementary Geometry by Alexander and Koeberlein 8.1 Example 1: (Finding the lateral area) The bases of the right prism shown are equilateral pentagons with sides of length 3 in. each. If the altitude of the prism is 4 in., …nd the lateral area of the prism. De…nition: "Total Area" kFor any prism, the total area T is the sum of the lateral area and the areas of the basesk Theorem 8.1.2: The total area T of any prism with lateral area L and base area B is given by Example 2: (Finding the total area) For a right triangular prism, suppose that the sides of the triangular base measure 3 f t; 4 f t; and 5 f t. The altitude is 6 f t: Find the total area. De…nition: "Regular Prism" kA regular prism is a right prims whose bases are regular polygonsk Example 3: (Finding the total area) Find the total area of the regular hexagonal prism with sides of length 4 in: and altitude 10 in: Page: 2 Bibiana Lopez Elementary Geometry by Alexander and Koeberlein De…nition: 8.1 "Cube" kA cube is a right square prims whose edges are congruentk Example 4: The total area of the right square prism is 210 cm2 . Find the length of a side of the square base if the altitude of the prism is 8 cm: Volume of a Prism Postulate 24: Volume Postulate kCorresponding to every solid is a unique positive number V known as the volume of that solidk Postulate 25: The volume of a right rectangular prism is given by where l measures the length, w is the width, and h is the altitude of the prism. Example 5: (Finding the volume) Find the volume of a box whose dimensions are 2 f t; 16 in; and 20 in: Postulate 26: The volume of a right prism is given by where B is the area of a base and h is the altitude of the prism. Page: 3 Bibiana Lopez Elementary Geometry by Alexander and Koeberlein 8.1 Example 6: (Finding the volume) For a right triangular prism, suppose that the sides of the triangular base measure 3 f t; 4 f t; and 5 f t, the altitude is 6 f t: Find the volume. Example 7: A concrete pad 4 in. thick is to have a length of 36 ft. and a width of 30 ft. How many cubic yards of concrete must be poured? [Hint: 1 yd3 = 27 f t3 ] Page: 4 Bibiana Lopez