# 9.25: Surface Area and Volume of Spheres

- Page ID
- 6748

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.

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.**

#### 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:** \(SA=4\pi r^{2}\)

## 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:** \(V=\dfrac{4}{3}\pi r^{3}\)

Example \(\PageIndex{1}\)

Find the surface area of the figure below, a hemisphere with a circular base.

**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[3]{ }, or you can use \(3375^{\dfrac{1}{3}}\).

\(\sqrt[3]{3375}=15\approx r\)

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

- Are there any cross-sections of a sphere that are not a circle? Explain your answer.
- List all the parts of a sphere that are the
as a circle.*same* - 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 \(\pi\).

- a radius of 8 in.
- a diameter of 18 cm.
- a radius of 20 ft.
- a diameter of 4 m.
- a radius of 15 ft.
- a diameter of 32 in.
- a circumference of \(26\pi \text{ cm}\).
- a circumference of \(50\pi\text{ yds}\).
- The surface area of a sphere is \(121\pi \text{ in}^{2}\). What is the radius?
- The volume of a sphere is \(47916\pi \text{ m}^{3}\). What is the radius?
- The surface area of a sphere is \(4\pi \text{ ft}^{2}\). What is the volume?
- The volume of a sphere is \(36\pi \text{ mi}^{3}\). What is the surface area?
- Find the radius of the sphere that has a volume of \(335 \text{ cm}^{3}\). Round your answer to the nearest hundredth.
- 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 .

## Review (Answers)

To see the Review answers, open this PDF file and look for section 11.7.

## 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. |

## Additional Resources

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!