Geometry

Surface Area and Volume of Spheres

April 20, 2023

Goals

Find the surface area of spheres. Find the volume of spheres. Solve problems using area and volume.

April 20, 2023

Sphere

radius

Great Circle

(Divides the sphere into two halves.)

Sphere Demo

The set of points in space that are equidistant from the same point, the center.

April 20, 2023

Hemisphere

r

Half of a sphere.

April 20, 2023

Sphere Formulas

r

2

343

4SA r

V r

April 20, 2023

Using a Calculator

You may find it easier to use the formula for volume in this form:

343rV

April 20, 2023

Example

2

2

2

343

343

323

4

4 2

16

50.3

2

33.5

SA r

V r

Find the Surface Area and the Volume of a sphere with a radius of 2.

April 20, 2023

Your Turn Find the surface area and volume.

7 in.

2

343

4 7

196

615.8

7

1372

31436.8

SA

V

April 20, 2023

Problem 1

A snowman is made with three spheres. The largest has a diameter of 24 inches, the next largest has a diameter of 20 inches, and the smallest has a diameter of 16 inches. Find the volume of the snowman.

April 20, 2023

Problem 1 Solution

A snowman is made with three spheres. The largest has a diameter of 24 inches, the next largest has a diameter of 20 inches, and the smallest has a diameter of 16 inches. Find the volume of the snowman.

The radii are 12 in, 10 in, and 8 in.

April 20, 2023

Problem 1 Solution

The radii are 12 in, 10 in, and 8 in.

343

343

343

8 2144.66

10 4188.79

12 7238.23

V

V

V

13,571.68 cu in

April 20, 2023

Or, for easier calculation…

3 3 34 4 43 3 3

3 3 343

43

(8 ) (10 ) (12 )

(8 10 12 )

(3240)

13,571.68

V

April 20, 2023

Problem 2This is a grain silo, as found on many farms. They are used to store feed grain and other materials. They are usually cylindrical with a hemispherical top.

Assume that the concrete part has a height of 50 feet, and the diameter of the cylinder is 18 feet.

Find the volume of the silo.

April 20, 2023

Problem 2 Solution

50

18 Volume of Cylinder

V = r2h

V = (92)(50)

V = 81 50

V = 4050

V 12723.5 cu. ft.

9

April 20, 2023

Problem 2 SolutionVolume of Hemisphere

343 9

972

3053.6 cu. ft.

V

50

18

9This is the volume of a sphere. The volume of the hemisphere is half of this value, which is 1526.8 cu. ft.

April 20, 2023

Problem 2 SolutionVolume of Cylinder

12723.5

Volume of Hemisphere

1526.8

Total Volume

12723.5 + 1526.8 = 14250.3 cu. ft.

50

18

9

April 20, 2023

Problem 2 ExtensionTotal Volume =14250.3 cu. ft.

One bushel contains 1.244 cubic feet. How many

bushels are in the silo?

14250.3 1.244 =

11455.2 bushels

April 20, 2023

Problem 3A sphere is inscribed inside a cube which measures 6 in. on a side. What is the ratio of the volume of the sphere to the volume of the cube?

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April 20, 2023

Problem 3 Solution

6

6

6

Volume of the Cube:

6 6 6 = 216 cu in

Radius of the Sphere:

3 in.

Volume of the Sphere:3

343

43

3

(27)

36

V

April 20, 2023

Problem 3 Solution

6

6

6

Volume of the Cube:

216 cu in

Volume of the Sphere:

36

Ratio of Volume of Sphere to Volume of Cube3

36 36

216 36 6 6

April 20, 2023

Problem 4A mad scientist makes a potionin a full spherical flask which has a diameter of 4 inches. To drink it, he pours it into a cylindrical cup with a diameter of 3.5 inches and is 3.5 inches high. Will the potion fit into the cup? If not, how much is left in the flask?

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April 20, 2023

Problem 4 Solution

Flask Volume:

Diameter = 4 inches

Radius = 2 inches

343 2

4 8

333.5 cu in

V

April 20, 2023

Problem 4 Solution

Cup Volume:

Diameter = 3.5 inches

Radius = 1.75 inches

Height = 3.5 inches

2(1.75 )(3.5)

33.7 cu in

V

33.5 cu in

April 20, 2023

Problem 4 Solution

The flask holds 33.5 cu in.

The cup holds 33.7 cu in.

Yes, the potion fits into the cup.

33.5 cu in

33.7 cu in

April 20, 2023

Problem 5

The surface area of a sphere is 300 cm2.

Find:

1. Its radius,

2. The circumference of a great circle,

3. Its volume.

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April 20, 2023

Problem 5 Solutions2

2

2

4

300 4

300 23.874

4

2

. 9

7

8

3.8

SA r

r

r

r

r

343

343

2

30.72

489.

2 (4.89)

.

8

(4 89 )

C r

V r

April 20, 2023

Summary

A sphere is the set of points in space equidistant from a center.

A hemisphere is half of a sphere. A great circle is the largest circle that can

be drawn on a sphere. The diameter of a great circle equals the diameter of the sphere.

April 20, 2023

Formulas to Know

r

2

343

4SA r

V r

April 20, 2023

Last ProblemOn a far off planet, Zenu was examining his next target, Earth. The radius of the Earth is 3963 miles. What is the volume of material that will be blown into space?

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April 20, 2023

Last Problem Solution

343

11

3963

2.607 10 cubic miles

260,711,883,000 cu mi

V

April 20, 2023

Practice Problems