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 m^{2}, cm^{2}, and mm^{2}.

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

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

= 4πr^{2 }

Thus the surface area of a sphere = 4πr^{2 }

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) = 4πr^{2}, here π = 22/7 = 3.141, r = 5 in ∴ SA = 4 × 3.141 × 5^{2} = 314.1 in^{2}

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) = πd^{2}, here π = 22/7 = 3.141, d= 9 cm ∴ SA = 3.141 × 9^{2} = 254.42 cm^{2}

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 cm^{2}