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, 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€r^{2 }

Thus the surface area of a sphere = 4Ï€r^{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) = 4Ï€r^{2}, hereÂ Ï€Â = 22/7 = 3.141,Â rÂ = 5 in âˆ´ SA = 4 Ã— 3.141 Ã— 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) = Ï€d^{2}, here Ï€ = 22/7 = 3.141, d= 2 Ã— 9 cm = 18 cm âˆ´ SA = 3.141 Ã— (18)^{2} = 1017.684 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) = ${\dfrac{C^{2}}{\pi }}$ , hereÂ C = 36 cm, Ï€Â = 22/7 = 3.141 âˆ´ SA = (36)^{2} Ã· 3.141 = 412.6 cm^{2}

Very useful