Learn to find the volume and surface area of spheres.

Course 3

Spheres

Vocabulary

sphere

hemisphere

great circle

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Course 3

Spheres

A sphere is the set of points in three dimensions that are a fixed distance from a given point, the center. A plane that intersects a sphere through its center divides the two halves or hemispheres. The edge of a hemisphere is a great circle.

Course 3

Spheres

Theorem 12.11: Surface Area of a Sphere

The surface area of a sphere with radius r is S = 4r2.

Ex. 1: Finding the Surface Area of a Sphere

Find the surface area. When the radius doubles, does the surface area double?

S = 4r2

= 422

= 16 in.2

S = 4r2

= 442

= 64 in.2

The surface area of the sphere in part (b) is four times greater than the surface area of the sphere in part (a) because 16 • 4 = 64

So, when the radius of a sphere doubles, the surface area DOES NOT double.

More . . .

If a plane intersects a sphere, the intersection is either a single point or a circle. If the plane contains the center of the sphere, then the intersection is a great circle of the sphere. Every great circle of a sphere separates a sphere into two congruent halves called hemispheres.

Ex. 2: Using a Great Circle

The circumference of a great circle of a sphere is 13.8 feet. What is the surface area of the sphere?

Solution:

Begin by finding the radius of the sphere.

C = 2r13.8 = 2r13.8 2r 6.9 = r

= r

Solution:

Using a radius of 6.9 feet, the surface area is:

S = 4r2 = 4(6.9)2 = 190.44 ft.2

So, the surface area of the sphere is 190.44 ft²

Theorem 12.12: Volume of a Sphere

The volume of a sphere with radius r is

V = 4r3.3

Additional Example 1: Finding the Volume of a Sphere

Find the volume of a sphere with radius 9 cm, both in terms of p and to the nearest tenth of a unit.

= 972p cm3 3,052.1 cm3

Volume of a sphere

Substitute 9 for r.

43

V = pr3

= p(9)343

Course 3

Spheres

Try This: Example 1

Find the radius of a sphere given volume of a sphere is 113.1 m³.

Find the circumference of the great circle given volume = 72p in³

Volume of a sphere

Course 3

Spheres

43

V = pr3

Lesson Quiz: Part 1

Find the volume of each sphere, both in terms of and to the nearest tenth. Use 3.14 for p.

1. r = 4 ft

2. d = 6 m

Find the surface area of each sphere, both in terms of and to the nearest tenth. Use 3.14 for p.

36 p m3, 113.0 m3

85.3 p ft3, 267.8 ft3

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Course 3

Spheres

2.25p mi2, 7.1 mi2

1936p in2, 6079.0 in23. r = 22 in

4. d = 1.5 mi

5. A basketball has a circumference of 29 in. To the nearest cubic inch, what is its volume?

Lesson Quiz: Part 2

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Course 3

Spheres

412 in3