Section 12-6Surface Areas and Volumes of Spheres

Essential Questions

How do you find surface areas of spheres?

How do you find volumes of spheres?

Vocabulary

1. Great Circle:

2. Pole:

3. Hemisphere:

Vocabulary

1. Great Circle:

2. Pole:

3. Hemisphere:

A circle formed when a plane intersects a sphere and the circle has the same center as the sphere

Vocabulary

1. Great Circle:

2. Pole:

3. Hemisphere:

A circle formed when a plane intersects a sphere and the circle has the same center as the sphere

The endpoints of the diameter of a great circle

Vocabulary

1. Great Circle:

2. Pole:

3. Hemisphere:

A circle formed when a plane intersects a sphere and the circle has the same center as the sphere

The endpoints of the diameter of a great circle

One of the two congruent halves of a sphere created by a great circle

Formulas

Surface Area of a Sphere:

Volume of a Sphere:

Formulas

Surface Area of a Sphere: SA = 4π r2

Volume of a Sphere:

Formulas

Surface Area of a Sphere: SA = 4π r2

Volume of a Sphere: V = 43π r3

Example 1Find the surface area of the sphere.

Example 1Find the surface area of the sphere.

SA = 4π r2

Example 1Find the surface area of the sphere.

SA = 4π r2

SA = 4π (4.5)2

Example 1Find the surface area of the sphere.

SA = 4π r2

SA = 4π (4.5)2

SA ≈ 254.47 in2

Example 2Find the surface area of a hemisphere with a diameter of 8 mm to the nearest hundredth.

Example 2Find the surface area of a hemisphere with a diameter of 8 mm to the nearest hundredth.

SA = 12 i4π r

2 + π r2

Example 2Find the surface area of a hemisphere with a diameter of 8 mm to the nearest hundredth.

SA = 12 i4π r

2 + π r2 r = 12 d

Example 2Find the surface area of a hemisphere with a diameter of 8 mm to the nearest hundredth.

SA = 12 i4π r

2 + π r2 r = 12 d

r = 12 (8)

Example 2Find the surface area of a hemisphere with a diameter of 8 mm to the nearest hundredth.

SA = 12 i4π r

2 + π r2 r = 12 d

r = 12 (8)

r = 4 mm

Example 2Find the surface area of a hemisphere with a diameter of 8 mm to the nearest hundredth.

SA = 12 i4π r

2 + π r2 r = 12 d

r = 12 (8)

r = 4 mm

SA = 2π (4)2 + π (4)2

Example 2Find the surface area of a hemisphere with a diameter of 8 mm to the nearest hundredth.

SA = 12 i4π r

2 + π r2 r = 12 d

r = 12 (8)

r = 4 mm

SA = 2π (4)2 + π (4)2

SA ≈150.8 mm2

Example 3Find the surface area of a sphere if the

circumference of the great circle is 14π inches.

Example 3Find the surface area of a sphere if the

circumference of the great circle is 14π inches.

SA = 4π r2

Example 3Find the surface area of a sphere if the

circumference of the great circle is 14π inches.

SA = 4π r2 C = 2π r

Example 3Find the surface area of a sphere if the

circumference of the great circle is 14π inches.

SA = 4π r2 C = 2π r14π = 2π r

Example 3Find the surface area of a sphere if the

circumference of the great circle is 14π inches.

SA = 4π r2 C = 2π r14π = 2π r2π 2π

Example 3Find the surface area of a sphere if the

circumference of the great circle is 14π inches.

SA = 4π r2 C = 2π r14π = 2π r2π 2πr = 7 in.

Example 3Find the surface area of a sphere if the

circumference of the great circle is 14π inches.

SA = 4π r2 C = 2π r14π = 2π r2π 2πr = 7 in.

SA = 4π (7)2

Example 3Find the surface area of a sphere if the

circumference of the great circle is 14π inches.

SA = 4π r2 C = 2π r14π = 2π r2π 2πr = 7 in.

SA = 4π (7)2

SA ≈ 615.75 in2

Example 4Find the volume of the sphere rounded to the

nearest hundredth.

Example 4Find the volume of the sphere rounded to the

nearest hundredth.

V = 43π r3

Example 4Find the volume of the sphere rounded to the

nearest hundredth.

V = 43π r3

V = 43π (4.5)3

Example 4Find the volume of the sphere rounded to the

nearest hundredth.

V = 43π r3

V = 43π (4.5)3

V ≈ 381.70 in3

Example 5Find the volume of a hemisphere with a diameter

of 6 feet to the nearest hundredth.

Example 5Find the volume of a hemisphere with a diameter

of 6 feet to the nearest hundredth.

V = 12i43π r3

Example 5Find the volume of a hemisphere with a diameter

of 6 feet to the nearest hundredth.

V = 12i43π r3 r = 1

2 d

Example 5Find the volume of a hemisphere with a diameter

of 6 feet to the nearest hundredth.

V = 12i43π r3 r = 1

2 d

r = 12 (6)

Example 5Find the volume of a hemisphere with a diameter

of 6 feet to the nearest hundredth.

V = 12i43π r3 r = 1

2 d

r = 12 (6)

r = 3 ft

Example 5Find the volume of a hemisphere with a diameter

of 6 feet to the nearest hundredth.

V = 12i43π r3 r = 1

2 d

r = 12 (6)

r = 3 ftV = 23π (3)3

Example 5Find the volume of a hemisphere with a diameter

of 6 feet to the nearest hundredth.

V = 12i43π r3 r = 1

2 d

r = 12 (6)

r = 3 ftV = 23π (3)3

V ≈ 56.55 ft3

Example 6Find the volume of a sphere that has a great

circle with an area of 9π square feet.

Example 6Find the volume of a sphere that has a great

circle with an area of 9π square feet.

V = 43π r3

Example 6Find the volume of a sphere that has a great

circle with an area of 9π square feet.

V = 43π r3 A = π r2

Example 6Find the volume of a sphere that has a great

circle with an area of 9π square feet.

V = 43π r3 A = π r2

9π = π r2

Example 6Find the volume of a sphere that has a great

circle with an area of 9π square feet.

V = 43π r3 A = π r2

9π = π r2π π

Example 6Find the volume of a sphere that has a great

circle with an area of 9π square feet.

V = 43π r3 A = π r2

9π = π r2π π

9 = r2

Example 6Find the volume of a sphere that has a great

circle with an area of 9π square feet.

V = 43π r3 A = π r2

9π = π r2π π

r = 3 ft.9 = r2

Example 6Find the volume of a sphere that has a great

circle with an area of 9π square feet.

V = 43π r3 A = π r2

9π = π r2π π

r = 3 ft.9 = r2

V = 43π (3)3

Example 6Find the volume of a sphere that has a great

circle with an area of 9π square feet.

V = 43π r3 A = π r2

9π = π r2π π

r = 3 ft.9 = r2

V = 43π (3)3

V ≈113.10 ft3

Problem Set

Problem Set

p. 868 #1-25 odd

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