Learn on PengiIllustrative Mathematics, Grade 8Chapter 5: Functions and Volume

Lesson 5: Dimensions and Spheres

In this Grade 8 Illustrative Mathematics lesson, students explore how changing a single dimension — such as the height of a cylinder or the edge length of a rectangular prism — affects volume through proportional relationships. Students write and graph equations relating volume to one variable for prisms, cylinders, and cones, then analyze how doubling or halving that dimension scales the volume by the same factor. The lesson reinforces that these volume relationships are linear functions and proportional relationships.

Section 1

Estimating Hemisphere Volume by Averaging

Property

The volume of a hemisphere can be estimated by averaging the volumes of a circumscribed cylinder and an inscribed cone. For a hemisphere of radius $r$ , the cylinder and cone have height $h=r$ . This estimation gives the exact volume of a hemisphere.

V_{est\_hemisphere} = \frac{V_{cylinder} + V_{cone}}{2} = \frac{\pi r^3 + \frac{1}{3}\pi r^3}{2} = \frac{2}{3}\pi r^3

Section 2

Volume of a Sphere

Property

The volume $V$ of a sphere with radius $r$ is given by the formula:

V = \frac{4}{3}\pi r^3

Examples

Section 3

Relationship Between Cone and Cylinder Volumes

Property

A cone has exactly one-third the volume of a cylinder with the same base and height: $V_{cone} = \frac{1}{3} \cdot V_{cylinder}$

If cylinder volume is $V_{cylinder} = \pi r^2 h$ , then cone volume is $V_{cone} = \frac{1}{3}\pi r^2 h$

Section 4

Sphere Volume from Cylinder Relationship

Property

The volume of a sphere is $\frac{2}{3}$ times the volume of a cylinder that has the same diameter as the sphere and height equal to the diameter:

V_{sphere} = \frac{2}{3} \times V_{cylinder} = \frac{2}{3} \times \pi r^2 h

When $h = 2r$ (diameter), this becomes:

V_{sphere} = \frac{2}{3} \times \pi r^2 (2r) = \frac{4}{3}\pi r^3

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Chapter 5: Functions and Volume

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Lesson 1: Inputs and Outputs
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Lesson 2
Lesson 2: Representing and Interpreting Functions
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Lesson 3: Linear Functions and Rates of Change
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Lesson 4: Cylinders and Cones
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Lesson 5: Dimensions and Spheres
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Section 1

Estimating Hemisphere Volume by Averaging

Property

V_{est\_hemisphere} = \frac{V_{cylinder} + V_{cone}}{2} = \frac{\pi r^3 + \frac{1}{3}\pi r^3}{2} = \frac{2}{3}\pi r^3

Section 2

Volume of a Sphere

Property

The volume $V$ of a sphere with radius $r$ is given by the formula:

V = \frac{4}{3}\pi r^3

Examples

Section 3

Relationship Between Cone and Cylinder Volumes

Property

A cone has exactly one-third the volume of a cylinder with the same base and height: $V_{cone} = \frac{1}{3} \cdot V_{cylinder}$

If cylinder volume is $V_{cylinder} = \pi r^2 h$ , then cone volume is $V_{cone} = \frac{1}{3}\pi r^2 h$

Section 4

Sphere Volume from Cylinder Relationship

Property

The volume of a sphere is $\frac{2}{3}$ times the volume of a cylinder that has the same diameter as the sphere and height equal to the diameter:

V_{sphere} = \frac{2}{3} \times V_{cylinder} = \frac{2}{3} \times \pi r^2 h

When $h = 2r$ (diameter), this becomes:

V_{sphere} = \frac{2}{3} \times \pi r^2 (2r) = \frac{4}{3}\pi r^3

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 5: Functions and Volume

Back to Illustrative Mathematics, Grade 8

Lesson 1
Lesson 1: Inputs and Outputs
Learn Practice
Lesson 2
Lesson 2: Representing and Interpreting Functions
Learn Practice
Lesson 3
Lesson 3: Linear Functions and Rates of Change
Learn Practice
Lesson 4
Lesson 4: Cylinders and Cones
Learn Practice
Lesson 5Current
Lesson 5: Dimensions and Spheres
Learn Practice

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Estimating Hemisphere Volume by Averaging

Property

V_{est\_hemisphere} = \frac{V_{cylinder} + V_{cone}}{2} = \frac{\pi r^3 + \frac{1}{3}\pi r^3}{2} = \frac{2}{3}\pi r^3

Section 2

Volume of a Sphere

Property

The volume $V$ of a sphere with radius $r$ is given by the formula:

V = \frac{4}{3}\pi r^3

Examples

Section 3

Relationship Between Cone and Cylinder Volumes

Property

A cone has exactly one-third the volume of a cylinder with the same base and height: $V_{cone} = \frac{1}{3} \cdot V_{cylinder}$

If cylinder volume is $V_{cylinder} = \pi r^2 h$ , then cone volume is $V_{cone} = \frac{1}{3}\pi r^2 h$

Section 4

Sphere Volume from Cylinder Relationship

Property

The volume of a sphere is $\frac{2}{3}$ times the volume of a cylinder that has the same diameter as the sphere and height equal to the diameter:

V_{sphere} = \frac{2}{3} \times V_{cylinder} = \frac{2}{3} \times \pi r^2 h

When $h = 2r$ (diameter), this becomes:

V_{sphere} = \frac{2}{3} \times \pi r^2 (2r) = \frac{4}{3}\pi r^3

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 5: Functions and Volume

Back to Illustrative Mathematics, Grade 8

Lesson 1
Lesson 1: Inputs and Outputs
Learn Practice
Lesson 2
Lesson 2: Representing and Interpreting Functions
Learn Practice
Lesson 3
Lesson 3: Linear Functions and Rates of Change
Learn Practice
Lesson 4
Lesson 4: Cylinders and Cones
Learn Practice
Lesson 5Current
Lesson 5: Dimensions and Spheres
Learn Practice

Lesson 5: Dimensions and Spheres

Property

Property

Examples

Property

Property

Chapter 5: Functions and Volume

Lesson 1: Inputs and Outputs

Lesson 2: Representing and Interpreting Functions

Lesson 3: Linear Functions and Rates of Change

Lesson 4: Cylinders and Cones

Lesson 5: Dimensions and Spheres

Lesson overview

Property

Property

Examples

Property

Property

Chapter 5: Functions and Volume

Lesson 1: Inputs and Outputs

Lesson 2: Representing and Interpreting Functions

Lesson 3: Linear Functions and Rates of Change

Lesson 4: Cylinders and Cones

Lesson 5: Dimensions and Spheres

Lesson 1: Inputs and Outputs

Lesson 2: Representing and Interpreting Functions

Lesson 3: Linear Functions and Rates of Change

Lesson 4: Cylinders and Cones

Lesson 5: Dimensions and Spheres