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Sphere Surface Area

Sep 13, 2022
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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.” 

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. 

parallel

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 

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Find the surface area of a sphere 

Example 1: 

Find the surface area of the sphere. 

Example 1: 

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. 

Example 2:

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. 

Example 3: 

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, 

r . 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 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.
  • 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 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

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.

Comments:

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