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Aim #11: What is the volume of a pyramid? Do Now: Cube B = Base area 2h = height Cube height scaled down by a factor of 2 CC Geometry H Pyramid ___ = Base area ___ = height 2h B B B Volume of Cube = __________ How many total pyramids with height h would be needed to equal the volume of the cube?___ Therefore, one pyramid represents _____ of the volume of the cube. Volume of Pyramid = ___ ______ or _______ Suppose we wanted to calculate the volume of the cone shown below. -Since the base, A, is an irregular shape, we could compare this cone to a right rectangular pyramid that has the same base area A and height h (shown below). -What does Cavalieri‛s Principle say about the volume of the general cone compared to the volume of the right rectangular pyramid? What can we conclude about the formula to find the volume of a general cone? Volume of a general cone = 1 (area of the base x height) 3 Exercises 3 1. A pyramid has volume 24 in and height 6 in. Find the area of its base. 2. A cone fits inside a cylinder so that their bases are the same and their heights are the same, as shown in the diagram below. Calculate the volume that is inside the cylinder, but outside of the cone. Give an exact answer. 12 5 3 3. A square pyramid has a volume of 245 in . The height of the pyramid is 15 in. What is the area of the base of the pyramid? What is the length of one side of the base? 4. Find the volume of the rectangular pyramid shown. 4m 9m 6m 5. The right circular cone shown has a base with radius of 7. The slant height of the cone‛s lateral surface is . Find the volume of the cone, to the nearest tenth. 3 6. Gold has a density of 19.32 g/cm . If a square pyramid has a base edge length of 5 cm, height of 6 cm, and a mass of 942 g, is the pyramid made of solid gold? Recall that density can be calculated with the formula: density = mass volume 7. Sand falls from a conveyor belt and forms a pile on a flat surface in the shape of a cone. The diameter of the pile is approximately 12 ft. and the height is approximately 5 ft. Estimate the volume of the pile of sand to the nearest cubic foot. 8. The frustum of a pyramid is formed by cutting off the top part by a plane parallel to the base. The base of the pyramid and the cross-section where the cut is made are called the bases of the frustum. The distance between the planes containing the bases is called the height of the frustum. Find the volume of the frustum shown if the bases are squares of edge lengths 2 in. and 3 in., and the height of the frustum is 4 in. Name ______________________ CC Geometry H Date _________________ HW #11 1. What is the volume formula for a right circular cone with radius r and height h? 2. Identify the solid shown, and find its volume. 4 3 6 3. Find the volume of the right square pyramid shown. 4. Find the volume of the circular cone below. Round your answer to the nearest hundredth. 5. Find the volume of a pyramid whose base is a square with edge length 3 cm. and whose height is also 3 cm. 6. Suppose you fill a conical paper cup with a height of 6‛‛ with water. If all the water is then poured into a cylindrical cup with the same radius and same height as the conical paper cup, to what height will the water reach in the cylindrical cup? 7. Use the diagram to answer the questions that follow. a. Determine the volume of the cone shown. Give an exact answer. b. Could a cone that has a height of 33 and radius 12 be the dimensions of a similar cone? Why or why not? c. Calculate the volume of the cone described in part (b) in two ways. (Hint: Use the volume formula and the scaling principle for volume.) Mixed Review 1. Two jars of peanut butter by the same brand are sold in a grocery store. The first jar is twice the height of the second jar, but its diameter is one-half as much as the shorter jar. The taller jar costs $1.89, and the shorter jar costs $2.79. Which jar is the better buy? Justify your answer. 2. The diagram shows rectangle ABCD with diagonal BD. What is the exact perimeter of ABCD and to the nearest tenth? A B 14 30 D o C