Navigation

6-8 Grade 8 Unit 5 Lesson 20

Gradeband

K-5 Math •6-8 Math 9-12 Math

Course

Grade 6 Grade 7 •Grade 8

Unit

Unit 1 Unit 2 Unit 3 Unit 4 •Unit 5 Unit 6 Unit 7 Unit 8 Unit 9

Lesson

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5 Lesson 6 Lesson 7 Lesson 8 Lesson 9 Lesson 10 Lesson 11 Lesson 12 Lesson 13 Lesson 14 Lesson 15 Lesson 16 Lesson 17 Lesson 18 Lesson 19 •Lesson 20 Lesson 21 Lesson 22

K-5
6-8
9-12

Grade 6
Grade 7
Grade 8

Accelerated 6
Accelerated 7

Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7
Unit 8
Unit 9

Lesson 1
Lesson 2
Lesson 3
Lesson 4
Lesson 5
Lesson 6
Lesson 7
Lesson 8
Lesson 9
Lesson 10
Lesson 11
Lesson 12
Lesson 13
Lesson 14
Lesson 15
Lesson 16
Lesson 17
Lesson 18
Lesson 19
Lesson 20
Lesson 21
Lesson 22

Lesson | Loading...

Sign in to access protected content and invite other educators

Sign in using your existing Kendall Hunt account. If you don’t have one, create an educator account.

Curriculum

IM K-5 Math
IM 6-8 Math
IM 9-12 Math

Professional Learning

Illustrative Mathematics® has operated as an independent 501(c)3 non-profit organization since 2013.

IM® 6-8 Math v.360, an IM v.360 curriculum, is ©2024 by Illustrative Mathematics®, and licensed under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0).

IM 6-8 Math v.360, an IM v.360 curriculum, is a derivative of IM 6-8 Math.

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics®, and is ©2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0) . OUR's 6-8 Math Curriculum is available at https://openupresources.org/math-curriculum/ .

Adaptations and updates to IM 6-8 Math v.360 are ©2024 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution-Noncommercial 4.0 International License (CC BY-NC 4.0).

Adaptations to add additional English language learner supports are ©2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

The second set of English assessments (marked as set “B”) are ©2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0) .

Spanish translation of the “B” assessments are ©2020 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written permission of Illustrative Mathematics.

This site includes public domain images or openly licensed images that are copyrighted by their respective owners. Openly licensed images remain under the terms of their respective licenses. See the image attribution section for more information.

Settings

Language

Audience

family

Not all roles available for this page.

Assessment Access

Sign in to view assessments and invite other educators

Sign in using your existing Kendall Hunt account. If you don’t have one, create an educator account.

English
Español

teacher
student
family

Lesson 20

20.1 Warm-up 20.2 Activity 20.3 Activity Student Lesson Summary

Unit 5, Lesson 20

The Volume of a Sphere

Grade 8

20.1

Warm-up

Sketch a Sphere

Here is a method for quickly sketching a sphere:

Draw a circle.
Draw an oval in the middle whose edges touch the sphere.

<p>Two images. First, a circle. Second, a circle drawn with an oval in the middle whose edges tough the sphere. Half of the oval is dotted to indicate that it would be on the back of the sphere.</p>

Practice sketching some spheres. Sketch a few different sizes.
For each sketch, draw a radius and label it .

Are You Ready for More?

20.2

Activity

A Sphere in a Cylinder

Here are a cone, a sphere, and a cylinder that all have the same radii and heights.

An image with three shapes: a cone, a sphere and a cylinder.

The radius of the cylinder is 5 units. When necessary, express all answers in terms of .

What is the height of the cylinder?
What is the volume of the cylinder?
What is the volume of the cone?
What is the volume of the sphere? Explain your reasoning.

Are You Ready for More?

20.3

Activity

Spheres in Cylinders

A cone, a sphere, and a cylinder that all have the same radius and height are shown here.

An image with three shapes: a cone, a sphere and a cylinder.

Let the radius of the cylinder be units. When necessary, express answers in terms of .

What is the height of the cylinder in terms of ?
What is the volume of the cylinder in terms of ?
What is the volume of the cone in terms of ?
What is the volume of the sphere in terms of ?
The volume of the cone is the volume of a cylinder. The volume of the sphere is what fraction of the volume of the cylinder?

Are You Ready for More?

Student Lesson Summary

Think about a sphere with radius units that fits snugly inside a cylinder. The cylinder must then also have a radius of units and a height of units. Using what we have learned about volume, the cylinder has a volume of , which is equal to cubic units.

We know from an earlier lesson that the volume of a cone with the same base and height as a cylinder has of the volume. In this example, such a cone has a volume of , or cubic units.

Three figures. First, cone, radius, r, height 2 r. Second, sphere, radius, r. Third, cylinder, radius, r, height, 2 r.

If we filled the cone and sphere with water and then poured that water into the cylinder, the cylinder would be completely filled. That means the volume of the sphere and the volume of the cone add up to the volume of the cylinder. In other words, if is the volume of the sphere, then

This leads to the formula for the volume of the sphere,

Glossary

None