AMSCO'S Geometry. New York - Rye High School

Volume of a Prism 447The other prism has bases that are triangles, LMN and PQR. If LMis 16 centimeters and the length of the altitude to LM is 15 centimeters, then the1area of a base is 2(16)(15) or 120 square centimeters. If the height of this prismis 10 centimeters, then:Volume of the triangular prism 5 Bh5 120 3 105 1,200 cubic centimetersNote that for these two prisms, the areas of the bases are equal and theheights of the prisms are equal. Therefore, the volumes of the prisms are equal.This is true in general since volume is defined as the area of the base times theheight of the prism.The terms “base” and “height” are used in more than one way when describinga prism. For example, each of the congruent polygons in parallel planes is abase of the prism. The distance between the parallel planes is the height of theprism. In order to find the area of a base that is a triangle or a parallelogram, weuse the length of a base and the height of the triangle or parallelogram. Whenfinding the area of a lateral face that is a parallelogram, we use the length of thebase and the height of that parallelogram. Care must be taken in distinguishingto what line segments the words “base” and “height” refer.EXAMPLE 1SolutionThe bases of a right prism are ABC and ABC with D a point on CB,AD ' BC, AB 10 cm, AC 10 cm, BC 12 cm, AD 8 cm, and BB15 cm.Find the volume of the prism.12 cmSince this is a right prism, all of the lateral CBfaces are rectangles and the height of theD8 cmprism, AA, is the height of each face.10 cm 10 cm15 cmEach base is an isosceles triangle.AThe length of the base of the isosceles Ctriangle is BC 12 cm, and the lengthof the altitude to the base of the triangleAis AD 8 cm.1Area of ABC 2 (BC)(AD)1 2 (12)(8) 48Since the prism is a right prism, the height of the prism is BB 15.Volume of the prism 5 (area of a base)(height of the prism)5 (48)(15)5 720 cubic centimetersAnswerB