Learn on PengiReveal Math, AcceleratedUnit 12: Area, Surface Area, and Volume

Lesson 12-8: Solve Problems Involving Volume of Spheres

In this Grade 7 Reveal Math Accelerated lesson from Unit 12, students learn to calculate the volume of spheres using the formula V = 4/3πr³ and apply it to real-world problems. Students practice finding a sphere's radius from its circumference before computing volume, and compare volumes across different-sized spheres. The lesson builds on prior knowledge of cone volume to derive the sphere volume formula and reinforces unit tracking throughout multi-step calculations.

Section 1

Volume of a Sphere

Property

The volume of a sphere is given by

\text{Volume} = \dfrac{4}{3} \times \pi r^3

where $r$ is the radius of the sphere. Recall that $r^3$ , which we read as ' $r$ cubed,' means $r \times r \times r$ .

Examples

A gumball has a radius of 1 centimeter. Its volume is $V = \frac{4}{3} \pi (1)^3 = \frac{4}{3}\pi \approx 4.19$ cubic centimeters.
A soccer ball has a diameter of 22 cm, so its radius is 11 cm. Its volume is $V = \frac{4}{3} \pi (11)^3 = \frac{4}{3} \pi (1331) \approx 5575.28$ cubic centimeters.
A spherical ornament has a volume of $36\pi$ cubic inches. To find its radius, solve $36\pi = \frac{4}{3}\pi r^3$ , which simplifies to $27 = r^3$ , so the radius is $r = \sqrt[3]{27} = 3$ inches.

Explanation

Volume measures the space inside a 3D shape, like a ball or a planet. For a sphere, you cube the radius (multiply it by itself three times), then multiply by pi ( $\pi$ ), and finally multiply by the fraction $\frac{4}{3}$ .

Section 2

Volume of Hemispheres

Property

The volume of a hemisphere is half the volume of a complete sphere:

V_{hemisphere} = \frac{1}{2} \cdot \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3

Examples

Section 3

Volume of a Sphere and Cone

Property

The volume of a sphere is exactly twice the volume of a cone that has the same radius ( $r$ ) and a height ( $h$ ) equal to the sphere's diameter ( $2r$ ).

V_{\text{sphere}} = 2 \cdot V_{\text{cone}} \quad \text{when } h = 2r

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Volume of a Sphere

Property

The volume of a sphere is given by

\text{Volume} = \dfrac{4}{3} \times \pi r^3

where $r$ is the radius of the sphere. Recall that $r^3$ , which we read as ' $r$ cubed,' means $r \times r \times r$ .

Examples

A gumball has a radius of 1 centimeter. Its volume is $V = \frac{4}{3} \pi (1)^3 = \frac{4}{3}\pi \approx 4.19$ cubic centimeters.
A soccer ball has a diameter of 22 cm, so its radius is 11 cm. Its volume is $V = \frac{4}{3} \pi (11)^3 = \frac{4}{3} \pi (1331) \approx 5575.28$ cubic centimeters.
A spherical ornament has a volume of $36\pi$ cubic inches. To find its radius, solve $36\pi = \frac{4}{3}\pi r^3$ , which simplifies to $27 = r^3$ , so the radius is $r = \sqrt[3]{27} = 3$ inches.

Explanation

Section 2

Volume of Hemispheres

Property

The volume of a hemisphere is half the volume of a complete sphere:

V_{hemisphere} = \frac{1}{2} \cdot \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3

Examples

Section 3

Volume of a Sphere and Cone

Property

The volume of a sphere is exactly twice the volume of a cone that has the same radius ( $r$ ) and a height ( $h$ ) equal to the sphere's diameter ( $2r$ ).

V_{\text{sphere}} = 2 \cdot V_{\text{cone}} \quad \text{when } h = 2r

Overview

Lesson overview

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

Overview

Section 1

Volume of a Sphere

Property

The volume of a sphere is given by

\text{Volume} = \dfrac{4}{3} \times \pi r^3

where $r$ is the radius of the sphere. Recall that $r^3$ , which we read as ' $r$ cubed,' means $r \times r \times r$ .

Examples

A gumball has a radius of 1 centimeter. Its volume is $V = \frac{4}{3} \pi (1)^3 = \frac{4}{3}\pi \approx 4.19$ cubic centimeters.
A soccer ball has a diameter of 22 cm, so its radius is 11 cm. Its volume is $V = \frac{4}{3} \pi (11)^3 = \frac{4}{3} \pi (1331) \approx 5575.28$ cubic centimeters.
A spherical ornament has a volume of $36\pi$ cubic inches. To find its radius, solve $36\pi = \frac{4}{3}\pi r^3$ , which simplifies to $27 = r^3$ , so the radius is $r = \sqrt[3]{27} = 3$ inches.

Explanation

Section 2

Volume of Hemispheres

Property

The volume of a hemisphere is half the volume of a complete sphere:

V_{hemisphere} = \frac{1}{2} \cdot \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3

Examples

Section 3

Volume of a Sphere and Cone

Property

The volume of a sphere is exactly twice the volume of a cone that has the same radius ( $r$ ) and a height ( $h$ ) equal to the sphere's diameter ( $2r$ ).

V_{\text{sphere}} = 2 \cdot V_{\text{cone}} \quad \text{when } h = 2r