# Surface Area of a Sphere

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:

Let us derive the formula of surface area.

#### Derivation

According to Archimedes, if a sphere and a cylinder have equal radius, â€˜râ€™, then,

Surface area of a sphere = Lateral surface area of a cylinder

Thus,

Lateral surface area of the cylinder = 2Ï€rh, here r = radius, h = height

Assuming the sphere perfectly fit within the cylinder

Height (h) of the cylinder = diameter of the sphere = 2r â€¦â€¦â€¦â€¦. (1)

âˆ´Lateral surface area of the cylinder = 2Ï€rh

= 2Ï€r Ã— (2r) â€¦â€¦â€¦â€¦from (1), âˆµ h = 2r

= 4Ï€r2

Thus the surface area of a sphere = 4Ï€r2

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) = 4Ï€r2, hereÂ Ï€Â = 22/7 = 3.141,Â rÂ = 5 in
âˆ´ SA = 4 Ã— 3.141 Ã— 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:

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) = Ï€d2, here Ï€ = 22/7 = 3.141, d = 2 Ã— 9 cm = 18 cm
âˆ´ SA = 3.141 Ã— (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) = ${\dfrac{C^{2}}{\pi }}$ , hereÂ  C = 36 cm, Ï€Â = 22/7 = 3.141
âˆ´ SA = (36)2 Ã· 3.141
= 412.6 cm2