lesson 26 introduction m.8.24 understand volume …...guided nstruction think about 222 curriculum...

Introduction

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Think It Through

Lesson 26 Understand Volume of Cylinders, Cones, and Spheres

Lesson 26Understand Volume of Cylinders, Cones, and Spheres

You already know that the volume of a rectangular prism is equal to the area of the base of the prism times the height.

h

wℓ

Volume of a rectangular prism 5 Area of the base • height 5 (length • width) • height

You can find the volume of a cylinder in the same way.

The volume of a cylinder can be found by multiplying the area of the base of the cylinder by the height. But in the case of the cylinder, the base is a circle, not a rectangle.

Use the formula for the area of a circle to find the area of the base of the cylinder. The radius of the cylinder is r, so the area of the base is pr2.

Now we can find the volume of the cylinder.

Volume of a cylinder 5 Area of the base • height 5 (pr2) • h 5 pr2h

Think How can you find the volume of a cylinder?

Circle the information you use to find the area of the base of the cylinder.

How is finding the volume of a cylinder like finding the volume of a rectangular prism?

r

h

M.8.24

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The cone and the half sphere below are both shown inside a cylinder. The cylinders, the cone, and the half sphere all have the same circle as their base. They all have the same height, too.

h

r

h

r

The volume of the cone is 1 ·· 3 the

volume of the cylinder.

The volume of the half sphere

is 2 ·· 3 the volume of the cylinder.

You can use these relationships to determine the formula for the volume of a cone and the formula for the volume of a sphere.

Volume of Cone Volume of Sphere

r

hr

r

hr

Volume 5 1 ·· 3 • area of the base • height Volume 5 2 • volume of half sphere

5 1 ·· 3 • area of a circle • height 5 2 • 2 ·· 3 • area of the base • height

5 1 ·· 3 pr2h or pr2h ··· 3 5 4 ·· 3 • pr2 • r

5 4 ·· 3 pr3

Reflect1 How can you use the formula for the volume of a cylinder to remember the formulas for

the volume of a cone and the volume of a sphere?

Think What are the formulas for the volume of a cone and the volume of a sphere?

How many half spheres make a whole sphere?

Guided Instruction

Think About


Lesson 26


Using Volume Formulas

4 feet

3 feet

4 feet

3 feet

2 How is the base of the cylinder related to the base of the cone?

3 How are the heights of the cylinder and the cone related?

4 Find the volume of the cylinder. Write your answer in terms of p.

5 Find the volume of the cone. Write your answer in terms of p.

6 How does the volume of the cylinder compare to the volume of the cone?

7 Suppose you fill the cone with water and empty the water into the cylinder. How many times will you empty the water from the cone into the cylinder? Explain.

Let’s Explore the Idea You can compare the volumes of cylinders and cones. Use the pictures below to answer problems 2–7.

©Curriculum Associates, LLC Copying is not permitted. 223Lesson 26 Understand Volume of Cylinders, Cones, and Spheres

Let’s Talk About It Solve the problems below as a group.

8 The formula for the volume of a cylinder is V 5 pr2h. The formula can also be written

V 5 Bh. What does B represent?

9 Complete the equation below using your answers to problems 5 and 6 on the previous page.

Volume of the cone • 5 Volume of the cylinder

10 Explain how you can use the equation in problem 9 to write a formula for the volume of any cone using B to represent the area of the base of the cone.

11 Compare the volume formulas for spheres and cylinders. Why do you find r3 to find the volume of a sphere when you only find r2 to find the volume of a cylinder?

Try It Another Way

12 If the radius of a cylinder is doubled, will the volume be doubled? Explain.

Guided Practice

Connect


Lesson 26



Talk through these problems as a class and write your answers below.

13 Compare The water glasses below are filled to the same height and have the same radius. How many times could you fill Glass B to equal the amount water in Glass A? Explain your reasoning.

r r

h hh

Glass A Glass B

14 Analyze If the radius of a sphere is doubled, how does the volume change? Support your answer by finding the volume of a sphere with r 5 2a and comparing it to the volume of a sphere with r 5 a.

15 Explain Explain what pr2 and h represent in terms of the cylinder. Then explain how

this information is used to find the formula for the volume of a half sphere, 2 ·· 3 • pr2 • r.

Then explain how you use this formula to get the formula 4 ·· 3 pr3 for the volume of

a sphere.

h

r

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Independent Practice

Apply

Lesson 26



16 Put It Together Use what you have learned to answer the questions below.

Part A Describe the relationships among the volumes of the solids. Which solid has the greatest volume? Which solid has the second greatest volume? Do any of the solids have the same volumes? Explain your reasoning. (Represent volumes in terms of p.)

A B C

r

h

2r

h

r

3h

Part B Describe the relationships among the volumes of the solids. Which solid has the greatest volume? Which solid has the second greatest volume? Do any of the solids have the same volumes? Explain your reasoning. (Represent any volumes in terms of p.)

A

r

h

B

r

3h

r

3h

C

h

3r

D

lesson 26 introduction m.8.24 understand volume …...guided nstruction think about 222 curriculum...

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