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.
