Sphere Surface Area

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, 4πr 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 = 4πr².  

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 m³. 

Example 2: 

Find the surface area of the sphere. 

Solution: 

Use the formula for the surface area of a sphere. 

S = 4πr² 

   = 4π(2)² 

   = 16π 

   ≈ 50.3 

Surface area is 50.3 ft³. 

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 = 4πr² 

   = 4π(3)² 

   = 36π 

   ≈ 113.1 

Surface area is 113.1 cm³. 

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, 

2πr . Find r. 

2πr = 29*1/2

2πr = 59/2

r = 59/4π

Find the surface area. 
S = 4πr² 

   = 4π (59/4π)²

   =59²/4π

   ≈ 277.0 

The surface area of a basketball used in a professional game is 277.0 in² 

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.