# SURFACE AREA AND VOLUME OF SPHERES WORKSHEET

Question 1 :

(a) Find the surface area of the sphere shown below.

(b) When the radius doubles, does the surface area double ? Question 2 :

The circumference of a great circle of a sphere is 13.8π feet. What is the surface area of the sphere ? Question 3 :

Find the volume of the sphere shown below. Question 4 :

A baseball and its leather covering are shown. The baseball has a radius of about 1.45 inches. a. Estimate the amount of leather used to cover the baseball.

b. The surface of a baseball is sewn from two congruent shapes, each of which resembles two joined circles. How does this relate to the formula for the surface area of a sphere ?

Question 5 :

To make a steel ball bearing, a cylindrical slug is heated and pressed into a spherical shape with the same volume. Find the radius of the ball bearing below.   Solution (a) :

Formula for surface area of a sphere :

S  =  4πr2

Substitute 2 for r.

S  =  4π (22)

S  =  4π (4)

S  =  16π

The surface area of the sphere is 16π square inches.

Solution (b) :

When the radius doubles,

r  =  2 ⋅ 2

r  =  4 inches

Formula for surface area of a sphere :

S  =  4πr2

Substitute 4 for r.

S  =  4π (42)

S  =  4π (16)

S  =  64π  in2

Because 16π ⋅ 4  =  64π, the surface area of the sphere in part (b) is four times the surface area of the sphere in part (a).

So, when the radius of a sphere doubles, the surface area does not double.

Draw a sketch. Begin by finding the radius of the sphere.

Formula for circumference of a circle :

C  =  2πr

Substitute 13.8π for C.

13.8π  =  2πr

Divide each side by 2π.

6.9  =  r

Formula for surface area of a sphere :

S  =  4πr2

Substitute 6.9 for r.

S  =  4π(6.9)2

Simplify.

S  =  4π( 47.61)

Use calculator.

S  ≈  598  ft2

So, the surface area of the sphere is about 598 square feet. Formula for volume of a sphere :

V  =  4/3 ⋅ πr3

Substitute 22 for r.

V  =  4/3 ⋅ π(223)

Simplify.

V  =  4/3 ⋅ π(10648)

V  =  42592/3 ⋅ π

Use calculator.

V    44602 cm2

The volume of the sphere is about 44602 cubic cm. Solution (a) :

Because the radius r is about 1.45 inches, the surface area is as follows.

Formula for surface area of a sphere :

S  =  4πr2

Substitute 1.45 for r.

S  =  4π(1.452)

Simplify.

S  =  8.41π

Use calculator.

S  ≈  26.4  in2

So, the amount of leather used to cover the baseball is about 26.4 square inches.

Solution (b) :

Because the covering has two pieces, each resembling two joined circles, then the entire covering consists of four circles with radius r.

The area of a circle of radius r is

A  =  πr2

So, the area of the covering can be approximated by

4πr2

This is the same as the formula for the surface area of a sphere. To find the radius of the ball bearing, first we need to find the volume of the slug.

Use the formula for the volume of a cylinder.

V  =  πr2h

Substitute 1 for r and 2 for h.

V  =  π(1)2(2)

Simplify.

V  =  2π  cm3

To find the radius of the ball bearing, use the formula for the volume of a sphere and solve for r.

Formula for volume of sphere :

V  =  4/3 ⋅ πr3

Substitute 2π for V.

2π  =  4/3 ⋅ πr3

Multiply each side by 3.

6π  =  4πr3

Divide each side by 4π.

1.5  =  r3

Use a calculator to take the cube root.

1.14  ≈  r

So, the radius of the ball bearing is about 1.14 centimeters.

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