# 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, ‘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:

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 = 9 cm
SA = 3.141 × 92
= 254.42 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