# 9.25: Surface Area and Volume of Spheres

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

Derive and use the formula: volume equals $$\dfrac{4}{3}$$ times pi times the radius cubed.

## Spheres

Consider a solid figure consisting of the set of all points in three-dimensional space that are equidistant from a single point. How could you determine how much two-dimensional and three-dimensional space that figure occupies?

A sphere is the set of all points in three-dimensional space that are equidistant from a single point. The radius of a sphere has one endpoint on the sphere surface and the other endpoint at the center of that sphere. The diameter of a sphere must contain the center. Figure $$\PageIndex{1}$$

A great circle is the largest circular cross-section in a sphere. The circumference of a sphere is the circumference of a great circle. Every great circle divides a sphere into two congruent hemispheres. Figure $$\PageIndex{2}$$

#### Surface Area

Surface area is a two-dimensional measurement that is the total area of all surfaces that bound a solid. The basic unit of area is the square unit.

Surface Area of a Sphere: $$SA=4\pi r^{2}$$ Figure $$\PageIndex{3}$$

## Volume

To find the volume of any solid you must figure out how much space it occupies. The basic unit of volume is the cubic unit.

Volume of a Sphere: $$V=\dfrac{4}{3}\pi r^{3}$$ Figure $$\PageIndex{4}$$

Example $$\PageIndex{1}$$

Find the surface area of the figure below, a hemisphere with a circular base. Figure $$\PageIndex{5}$$

Solution

Use the formula for surface area:

\begin{aligned} SA&=\pi r^{2} +124\pi r^{2} \\ &=\pi (6^{2})+2\pi (6^{2}) \\&=36\pi +72\pi =108\pi \text{ cm}^{2}\end{aligned}

Example $$\PageIndex{2}$$

A sphere has a volume of 14,137.167 ft3. What is the radius?

Solution

Use the formula for volume, plug in the given volume and solve for the radius, r:

\begin{aligned} V&=\dfrac{4}{3}\pi r^{3} \\ 14,137.167&=\dfrac{4}{3}\pi r^{3} \\ \dfrac{3}{4\pi} \cdot 14,137.167&=r^{3} \\ 3375&\approx r^{3}\end{aligned}

At this point, you will need to take the cubed root of 3375. Your calculator might have a button that looks like \sqrt{ }, or you can use $$3375^{\dfrac{1}{3}}$$.

$$\sqrt{3375}=15\approx r$$

Example $$\PageIndex{3}$$

The circumference of a sphere is 26\pi feet. What is the radius of the sphere?

Solution

The circumference is referring to the circumference of a great circle.

Use $$C=2\pi r$$:

\begin{aligned} 2\pi r&=26\pi \\ r&=13\text{ ft}\end{aligned}

Example $$\PageIndex{4}$$

Find the surface area of a sphere with a radius of 14 feet.

Solution

Use the formula for surface area:

$$SA=4\pi (14)^{2}$$

$$=784\pi \text{ ft}^{2}$$\)

Example $$\PageIndex{5}$$

Find the volume of a sphere with a radius of 6 m.

Solution

Use the formula for volume:

\begin{aligned} V&=\dfrac{4}{3}\pi 63 \\ &=\dfrac{4}{3}\pi (216) \\ &=288\pi \text{ m}^{3}\end{aligned}

## Review

1. Are there any cross-sections of a sphere that are not a circle? Explain your answer.
2. List all the parts of a sphere that are the same as a circle.
3. List any parts of a sphere that a circle does not have.

For 4 - 11, find the surface area and volume of a sphere with the given dimension. Leave your answer in terms of $$\pi$$.

1. a radius of 8 in.
2. a diameter of 18 cm.
3. a radius of 20 ft.
4. a diameter of 4 m.
5. a radius of 15 ft.
6. a diameter of 32 in.
7. a circumference of $$26\pi \text{ cm}$$.
8. a circumference of $$50\pi\text{ yds}$$.
9. The surface area of a sphere is $$121\pi \text{ in}^{2}$$. What is the radius?
10. The volume of a sphere is $$47916\pi \text{ m}^{3}$$. What is the radius?
11. The surface area of a sphere is $$4\pi \text{ ft}^{2}$$. What is the volume?
12. The volume of a sphere is $$36\pi \text{ mi}^{3}$$. What is the surface area?
13. Find the radius of the sphere that has a volume of $$335 \text{ cm}^{3}$$. Round your answer to the nearest hundredth.
14. Find the radius of the sphere that has a surface area $$225\pi \text{ ft}^{2}$$.

Find the surface area and volume of the following shape. Leave your answers in terms of \pi . Figure $$\PageIndex{6}$$

## Vocabulary

Term Definition
diameter A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.
Sphere A sphere is a round, three-dimensional solid. All points on the surface of a sphere are equidistant from the center of the sphere.
Radius The radius of a circle is the distance from the center of the circle to the edge of the circle.
Volume Volume is the amount of space inside the bounds of a three-dimensional object.
Cavalieri's Principle States that if two solids have the same height and the same cross-sectional area at every level, then they will have the same volume.

Interactive Element

Video: Spheres Principles - Basic

Activities: Spheres Discussion Questions

Study Aids: Spheres Study Guide

Practice: Surface Area and Volume of Spheres

Real World: Where We Live!

9.25: Surface Area and Volume of Spheres is shared under a CK-12 license and was authored, remixed, and/or curated by CK-12 Foundation via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.