What if you were given a solid figure consisting of the set of all points, in three-dimensional space, that are equidistant from a point? How could you determine how much two-dimensional and three-dimensional space that figure occupies? After completing this Concept, you'll be able to find the surface area and volume of a sphere.

### Watch This

### Guidance

A **sphere** is the set of all points, in three-dimensional space, which are equidistant from a point. The ** radius** has one endpoint on the sphere and the other endpoint at the center of that sphere. The

**of a sphere must contain the center.**

*diameter*

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. The best way to understand the surface area of a sphere is to watch the link by Russell Knightley, http://www.rkm.com.au/ANIMATIONS/animation-Sphere-Surface-Area-Derivation.html

**Surface Area of a Sphere:** .

##### 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. To see an animation of the volume of a sphere, see http://www.rkm.com.au/ANIMATIONS/animation-Sphere-Volume-Derivation.html by Russell Knightley.

**Volume of a Sphere:** .

#### Example A

The circumference of a sphere is . What is the radius of the sphere?

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

#### Example B

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

Use the formula.

#### Example C

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

Use the formula for volume:

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### Guided Practice

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

2. The surface area of a sphere is . What is the radius?

3. A sphere has a volume of . What is the radius?

**Answers:**

1. Use the formula for surface area.

2. Use the formula for surface area.

3. Use the formula for volume, plug in the given volume and solve for the radius, .

At this point, you will need to take the ** cubed root** of 3375. Your calculator might have a button that looks like , or you can do .

.

### Explore More

- 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.

Find the surface area and volume of a sphere with: (Leave your answer in terms of )

- 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 .
- a circumference of .
- The surface area of a sphere is . What is the radius?
- The volume of a sphere is . What is the radius?
- The surface area of a sphere is . What is the volume?
- The volume of a sphere is . What is the surface area?
- Find the radius of the sphere that has a volume of . Round your answer to the nearest hundredth.
- Find the radius of the sphere that has a surface area .

Find the surface area and volume of the following shape. Leave your answers in terms of .

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 11.7.