Volumes of Pyramids & Cones Lesson 12.5

Find the volume of the pyramid. V = 1/3Bh V = 1/3(15)(15)(12) Theorem 119: The volume of a pyramid is equal to one third of the product of the height and the area of the base. V = 1/3Bh Find the volume of the pyramid. V = 1/3Bh V = 1/3(15)(15)(12) V = 900 units3 12 15

Theorem 120: The volume of a cone is equal to one third of the product of the height and the area of the base. V = 1/3Bh = 1/3πr2h Find the volume of the cone. V = 1/3πr2h V = 1/3π(62)(9) V = 1/3π(36)(9) V = 108π units3

Cross Section of Pyramid or Cone Not congruent to the figure’s base. Is parallel and similar to the figure’s base.

Theorem 121: In a pyramid or cone, the ratio of the area of the cross section to the area of the base equals the square of the ratio of the figures’ respective distances from the vertex. C = k2 B h2 Where C is the area of the cross section, B is the area of the base, k is the distance from the vertex to the cross section, and h is the height of the pyramid or cone.

V = 1/3Bh B = s2/4 (√3) = 16√3 V = 1/3(16√3 )(21) V = 112√3 If the height of a pyramid is 21 and the pyramid’s base is an equilateral triangle with sides measuring 8, what is the pyramid’s volume? V = 1/3Bh B = s2/4 (√3) = 16√3 V = 1/3(16√3 )(21) V = 112√3

Vupper pyramid = 1/3(27/2)(9) Vupper pyramid = 40.5 cm3 A pyramid has a base area of 24 cm2 and a height of 12 cm. A cross section is cut 3 cm from the base. Find the volume of the upper pyramid (the solid above the cross section) Since the cross section is 3 cm from the base, its distance, k, from the peak is 9cm. C = k2 B h2 C = 92 24 122 C= 27/2 Vupper pyramid = 1/3Ck Vupper pyramid = 1/3(27/2)(9) Vupper pyramid = 40.5 cm3