The Volume Formula of a Pyramid and Cone

New York State Common Core Math Geometry, Module 3, Lesson 11

Worksheets for Geometry

Student Outcomes

• Students use Cavalieri’s principle and the cone cross section theorem to show that a general pyramid or cone has volume 1/3Bh where B is the area of the base and h is the height by comparing it with a right rectangular pyramid with base area B and height h.

The Volume Formula of a Pyramid and Cone

Classwork

Exploratory Challenge

Use the provided manipulatives to aid you in answering the questions below.
a. i. What is the formula to find the area of a triangle?
ii. Explain why the formula works.
b. i. What is the formula to find the volume of a triangular prism?
ii. Explain why the formula works.
c. i. What is the formula to find the volume of a cone or pyramid?
ii. Explain why the formula works

• Show Video

Exercises

1. 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.
2. A square pyramid has a volume of 245 in³. 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?
3. Use the diagram below to answer the questions that follow.
   a. Determine the volume of the cone shown below. Give an exact answer.
   b. Find the dimensions of a cone that is similar to the one given above. Explain how you found your answers.

Exercises

1. 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.
2. A square pyramid has a volume of 245 in³. 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?
3. Use the diagram below to answer the questions that follow.
   a. Determine the volume of the cone shown below. Give an exact answer.
   b. Find the dimensions of a cone that is similar to the one given above. Explain how you found your answers.
   c. Calculate the volume of the cone that you described in part (b) in two ways. (Hint: Use the volume formula and the scaling principle for volume.)
4. Gold has a density of 19.32 g/cm³. 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 in fact solid gold? If it is not, what reasons could explain why it is not? Recall that density can be calculated with the formula density = mass/volume.

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