**Surface Area and Volume of Spheres Worksheet :**

Worksheet given in this section is much useful to the students who would like to practice problems on surface area and volume of spheres.

Before look at the worksheet, if you would like to know the basic stuff about surface area and volume of spheres,

**Problem 1 :**

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

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

**Problem 2 :**

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

**Problem 3 : **

Find the volume of the sphere shown below.

**Problem 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 ?

**Problem 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.

**Problem 1 :**

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

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

**Solution :**

**Solution (a) : **

Formula for surface area of a sphere :

S = 4πr^{2}

Substitute 2 for r.

S = 4π (2^{2})

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 = 4πr^{2}

Substitute 4 for r.

S = 4π (4^{2})

S = 4π (16)

S = 64π in^{2}

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.

**Problem 2 :**

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

**Solution : **

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 = 4πr^{2}

Substitute 6.9 for r.

S = 4π(6.9)^{2}

Simplify.

S = 4π( 47.61)

Use calculator.

S ≈ 598 ft^{2}

So, the surface area of the sphere is about 598 square feet.

**Problem 3 : **

Find the volume of the sphere shown below.

**Solution :**

Formula for volume of a sphere :

V = 4/3 ⋅ πr^{3}

Substitute 22 for r.

V = 4/3 ⋅ π(22^{3})

Simplify.

V = 4/3 ⋅ π(10648)

V = 42592/3 ⋅ π

Use calculator.

V ≈ 44602 cm^{2}

The volume of the sphere is about 44602 cubic cm.

**Problem 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 ?

**Solution : **

**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 = 4πr^{2}

Substitute 1.45 for r.

S = 4π(1.45^{2})

Simplify.

S = 8.41π

Use calculator.

S ≈ 26.4 in^{2}

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 = πr^{2}

So, the area of the covering can be approximated by

4πr^{2}

This is the same as the formula for the surface area of a sphere.

**Problem 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 :**

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 = πr^{2}h

Substitute 1 for r and 2 for h.

V = π(1)^{2}(2)

Simplify.

V = 2π cm^{3}

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 ⋅ πr^{3}

Substitute 2π for V.

2π = 4/3 ⋅ πr^{3}

Multiply each side by 3.

6π = 4πr^{3}

Divide each side by 4π.

1.5 = r^{3}

Use a calculator to take the cube root.

1.14 ≈ r

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

After having gone through the stuff given above, we hope that the students would have understood, "Surface Area and Volume of Spheres Worksheet".

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