Surface Area of a Sphere – Explanation & Examples

The sphere is one of the important 3d figures in geometry. To recall, a sphere is a 3-dimensional object whereby every point is equidistance (same distance) from a fixed point, known as the sphere’s center. The diameter of a sphere divides it into two equal halves, called hemispheres.

The surface area of a sphere is the measure of the region covered by the surface of a sphere.

In this article, you will learn how to find the surface area of a sphere using the surface area of a sphere formula.

How to Find the Surface Area of a Sphere?

Like a circle, the distance from the center of a sphere to the surface is known as the radius. The surface area of a sphere is four times the area of the circle with the same radius.

Surface area of a sphere formula

The surface area of a sphere formula is given as:

Surface area of a sphere = 4πr² square units ……………. (Surface area of a sphere formula)

For a hemisphere (a half of a sphere), the surface area is given by;

Surface area of a hemisphere = ½ × surface area of sphere + area of the base (a circle)

= ½ × 4π r² + π r² 

Surface of a hemisphere = 3πr² …………………. (Surface area of a hemisphere formula)

Where r = the radius of the given sphere.

Let’s solve a few example problems about the surface area of a sphere.

Example 1

Calculate the surface area of a sphere of radius 14 cm.

Solution

Given:

Radius, r =14 cm

By the formula,

Surface area of a sphere = 4πr² 

On substitution, we get,

SA = 4 x 3.14 x 14 x 14

= 2,461.76 cm².

Example 2

The diameter of a baseball is 18 cm. Find the surface area of the ball.

Solution

Given,

Diameter = 18 cm ⇒ radius = 18/2 = 9 cm

A baseball has a spherical shape, therefore,

The surface area = 4πr² 

= 4 x 3.14 x 9 x 9

SA = 1,017.36 cm²

Example 3

The surface area of a spherical object is 379.94 m². What is the radius of the object?

Solution

Given,

SA = 379.94 m²

But, surface area of a sphere = 4πr² 

⇒ 379.94 = 4 x 3.14 x r²

⇒ 379.94 =12.56r²

Divide both sides by 12.56 and then find the square of the result

⇒ 379.94/12.56 = r²

⇒ 30.25 = r²

⇒ r = √30.25

= 5.5

Therefore, the radius of the spherical solid is 5.5 m.

Example 4

The cost of leather is $10 per square meter. Find the cost of manufacturing 1000 footballs of radius 0.12 m.

Solution

First, find the surface area of a ball

SA = 4πr² 

= 4 x 3.14 x 0.12 x 0.12

= 0.181 m²

The cost of manufacturing a ball = 0.181 m² x $10 per square meter

= $1.81

Therefore, the total cost of manufacturing 1000 balls = $1.81 x 1000

= $1,810

Example 5

The radius of the Earth is said to be 6,371 km. What is the surface area of the Earth?

Solution

The Earth is a sphere.

SA = 4πr² 

= 4 x 3.14 x 6,371 x 6,371

= 5.098 x 10⁸ km²

Example 6

Calculate the surface area of a solid hemisphere of radius 10 cm.

Solution

Given:

Radius, r = 10 cm

For a hemisphere, the surface area is given by:

SA = 3πr²

Substitute.

SA = 3 x 3.14 x 10 x 10

= 942 cm²

So, the surface area of the sphere is 942 cm².

Example 7

The surface area of a solid hemispherical object is 150.86 ft². What is the diameter of the hemisphere?

Solution

Given:

SA = 150.86 ft².

Surface area of a sphere = 3πr²

⇒ 150.86 = 3 x 3.14 x r²

⇒ 150.86 = 9.42 r²

Divide both sides by 9.42 to get,

⇒ 16.014 = r²          

r = √16.014

= 4

Hence, the radius is 4 ft, but the diameter is twice the radius.

So, the diameter of the hemisphere is 8 ft.

Example 8

Calculate the surface area of a sphere whose volume is 1,436.03 mm³.

Solution

Since, we already know that:

Volume of a sphere = 4/3 πr³

1,436.03 = 4/3 x 3.14 x r³

1,436.03 = 4.19 r³

Divide both sides by 4.19

r³ = 343

r = ³√343

r = 7

So, the radius of the sphere is 7 mm.

Now calculate the surface area of the sphere.

Surface area of a sphere = 4πr² 

= 4 x 3.14 x 7 x 7

= 615.44 mm².

Example 9

Calculate the surface area of a globe of radius 3.2 m

Solution

Surface area of a sphere
= 4π r²
= 4π (3.2)²
= 4 × 3.14 × 3.2 × 3.2
= 128.6 m²

Hence, the surface area of the globe is 128.6 m².