# Volumes of Cones and Cylinders

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Examples, videos, and solutions to help Grade 8 students learn the volume formulas for cones and cylinders.

### New York State Common Core Math Grade 8, Module 5, Lesson 10

Lesson 10 Student Outcomes

• Students know the volume formulas for cones and cylinders.
• Students apply the formulas for volume to real-world and mathematical problems.

Lesson 10 Summary

The formula to find the volume of a right circular cylinder is V = πr2h = Bh, where B is the area of the base. The formula to find the volume of a cone is directly related to that of the cylinder. Given a right circular cylinder with radius r and height h, the volume of a cone with those same dimensions is exactly one-third of the cylinder. The formula for the volume V, of a cone is V = 1/3 πr2h = 1/3 Bh, where B is the area of the base.

Lesson 10 Classwork

Exercises
1. a. - c. Write an equation to determine the volume of the rectangular prism shown below.
d. Write an equation for volume, V, in terms of the area of the base, B.

2. Using what you learned in Exercise 1, write an equation to determine the volume of the cylinder shown below.

3. - 5. Use the diagram at right to answer the questions.
a. What is the area of the base?
b. What is the height?
c. What is the volume of the rectangular prism?

Exercises 6–8
6. - 7. Use the diagram to find the volume of the right cone.

8. Challenge: A container in the shape of a right circular cone has height h, and base of radius r as shown. It is filled with water (in its upright position) to half the height. Assume that the surface of the water is parallel to the base of the inverted cone. Use the diagram to answer the following questions:
a. What do we know about the lengths of AB and AO?
b. What do we know about the measure of OAB and OCD?
c. What can you say about triangle OAB and triangle OCD?
d. What is the ratio of the volume of water to the volume of the container itself?

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