The surface area of a sphere is the entire region covered by its outer round surface. It is also the curved surface area of a sphere. Like all other surface area it is expressed in square units such as m2, cm2, and mm2.
We will learn how to find the surface area of a solid sphere. The equations are given below.
Formulas
The basic formula is given below:
With Radius
Surface Area of a Sphere
Let us derive the formula of surface area.
Derivation
According to Archimedes, if a sphere and a cylinder have equal radius, ‘r’, then,
Surface area of a sphere = Lateral surface area of a cylinder
Thus,
Lateral surface area of the cylinder = 2πrh, here r = radius, h = height
Assuming the sphere perfectly fit within the cylinder
Height (h) of the cylinder = diameter of the sphere = 2r …………. (1)
∴Lateral surface area of the cylinder = 2πrh
= 2πr × (2r) …………from (1), ∵ h = 2r
= 4πr2
Thus the surface area of a sphere = 4πr2
Let us solve an example involving the above formula.
Find the surface area of a sphere whose radius is 5 in.
Solution:
As we know, Surface Area (SA) = 4πr2, here π = 22/7 = 3.141, r = 5 in ∴ SA = 4 × 3.141 × 52 = 314.1 in2
Let us find the surface area of a sphere when the radius is not given directly.
With Diameter
The formula to find the surface area of a sphere using diameter is:
Surface Area of a Sphere With Diameter
Let us solve an example involving the above formula.
Find the surface area of a sphere with a radius of 9 cm.
Solution:
As we know, Surface Area (SA) = πd2, here π = 22/7 = 3.141, d= 2 × 9 cm = 18 cm ∴ SA = 3.141 × (18)2 = 1017.684 cm2
Let us find out the surface area of a sphere from circumference.
Finding the surface area of a sphere when the CIRCUMFERENCE is known
Find the surface area of a sphere with a circumference of 36 cm.
Solution:
Here we will use an alternative formula for the surface area using the circumference. Surface Area (SA) = ${\dfrac{C^{2}}{\pi }}$ , here C = 36 cm, π = 22/7 = 3.141 ∴ SA = (36)2 ÷ 3.141 = 412.6 cm2
Very useful