## Key Concepts

- Find the surface area of a sphere.
- Use the circumference of a sphere.

### Introduction

#### Sphere

A sphere is the locus of points in space that are a given distance from a point. The point is called the “center of the sphere.”

#### Radius

Radius is a segment from the center to a point on the sphere.

#### Chord

Chord of a sphere is a segment whose endpoints are on the sphere.

#### Diameter

Diameter of a chord that contains the center.

#### Great circle

If a plane intersects a sphere and contains the center, then the intersection is called a great circle.

#### Hemisphere

Every great circle of a sphere separates a sphere into two congruent halves called hemispheres.

#### Surface Area of a Sphere

Sphere Surface Area baseball can model a sphere. To approximate its surface area, you can take apart its covering. Each of the two pieces suggests a pair of circles with radius *r*, which is approximately the radius of the ball. The area of the four circles, 4π*r *^{2}, suggests the surface area of the ball.

If a sphere has a surface area of *SA *square units and a radius of *r *units, then

*SA *= 4π*r*^{2}.

**Surface Area = 4π (radius) ^{2}**

### Find the surface area of a sphere

**Example 1:**

Find the surface area of the sphere.

**Solution:**

Use the formula for the surface area of a sphere.

S = 4πr ^{2}

= 4π(9)^{2}

= 324π

≈ 1017.9

Surface area is 1017.9 m^{3}.

**Example 2:**

Find the surface area of the sphere.

**Solution:**

Use the formula for the surface area of a sphere.

S = 4πr^{2}

= 4π(2)^{2}

= 16π

≈ 50.3

Surface area is 50.3 ft^{3}.

### Use the circumference of a sphere

**Example 3:**

Find the surface area of the sphere.

**Solution:**

The diameter of the sphere is 6 cm, so the radius is

6/2 = 3 cm.

Use the formula for the surface area of a sphere.

S = 4πr^{2}

= 4π(3)^{2}

= 36π

≈ 113.1

Surface area is 113.1 cm^{3}.

**Example 4:**

Basketballs used in professional games must have a circumference of 29 ½ inches. What is the surface area of a basketball used in a professional game?

**Solution:**

We know that the circumference of a great circle is,

2π*r *. Find *r*.

2πr = 29*1/2

2πr = 59/2

r = 59/4π

Find the surface area.

S = 4πr^{2}

= 4π (59/4π)^{2}

=59^{2}/4π

≈ 277.0

The surface area of a basketball used in a professional game is 277.0 in^{2}

## Exercise

- A sphere is the locus of points in space that are a fixed distance from a given point called the ______________________.
- A _____________________ connects the center of the sphere to any point on the sphere.
- A ______________________ is half of a sphere.
- A _______________ divides a sphere into two hemispheres.
- Find the surface area of a sphere to the nearest tenth if the radius of the sphere is 6 cm.

- Find the surface area of the sphere. Round to the nearest tenth.

- Find the surface area of the sphere. Round to the nearest tenth.

- Nancy cuts a spherical orange in half along a great circle. If the radius of the orange is 2 inches, what is the area of the cross-section that Nancy cut? Round your answer to the nearest hundredth.
- The planet Saturn has several moons. These can be modeled accurately by spheres. Saturn’s largest moon Titan has a radius of about 2575 kilometers. What is the approximate surface area of Titan? Round your answer to the nearest tenth.

- The circumference of Earth is about 24,855 miles. Find the surface area of the Western Hemisphere of Earth.

### Concept Map

### What have we learned

- Find the surface area of a sphere using the surface formula.
- Use the circumference of a sphere to solve a problem.

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