Area of Sphere

From the smallest cricket ball to the largest celestial body, spheres are all around us and understanding the surfaces related to the sphere is important for solving many real-life applications. The area of a sphere is one such application. The total surface area of a sphere is the area covered by the sphere in a three-dimensional setting.

1.0Understanding the Sphere

The sphere is nothing but a three-dimensional circle in which all the points on the surface lie at equal distances from the centre. In other words, it is the group of certain points at the same distance from the centre. The distance, like a circle, from the centre to the outer surface is known as the radius of the sphere, and the line joining two opposite sides of the sphere passing through the centre is known as the diameter of the sphere. 

Diameter= $2\times Radius$

2.0What is the Formula for Sphere Area?

A sphere is the only three-dimensional figure having the same or equal curved and total surface area. The curved surface area of any 3-D figure is the area of the curved part only, while the total surface area includes the curved surface area along with the area of the base and/or the top part of the figure. 

$\text{Surface Area of the sphere(A) }=4(\pi )r^2$

“The unit for the surface area of the sphere is the same as the square of the sphere's radius while solving the question.” 

Area of a Sphere in Diameter

If, while solving any question, the diameter (d) of the sphere is given instead of the radius (r) of the sphere, then the formula for the area of the sphere can be expressed by using the relationship, $r=\frac{d}{2}$ , as: 

$\text{Surface Area of the sphere(A)}=4\pi {\left(\frac{d}{2}\right)}^2$

$A=4\pi \times \frac{d^2}{4}=\pi d^2$

Steps to Use the Formula

Step 1: Determine the radius of the sphere and convert the unit into the required unit if needed. 

Step 2: Solve the question using the correct values of each component. Always use $\pi =\frac{22}{7}$ if not mentioned otherwise. 

3.0The Surface Area of the Sphere Proof

We can find the area of a sphere derivation by comparing the sphere with the lateral or curved surface area of a cylinder, which is also a 3-D figure. The derivation for the formula is:

Step 1: Comparing the sphere to a cylinder

Consider a sphere fitting perfectly within a cylinder with the same radius(r) as the cylinder and height (h) of the cylinder with the same as the diameter of the sphere such that h = 2r 

Step 2: Lateral surface area of a cylinder

Lateral surface area of cylinder = $2(\pi )rh$

We know h = 2r

So, Lateral surface area of cylinder = $2(\pi )r(2r)$

The new lateral surface area of a cylinder = $4(\pi )r^2$

Since the surface area of the sphere = the lateral surface area of the cylinder 

Hence, 

The surface area of the sphere = $4(\pi )r^2$

4.0Solved Examples

Problem 1: The surface area of a spherical tank is 1540 cm². Find the radius of the tank.

Solution: It is given that A = 1540cm² 

$\text{Surface Area of the sphere(A)}=4(\pi )r^2$

$1540=4\times \frac{22}{7}\times r^2$

$r^2=\frac{7\times 1540}{22\times 4}$

$r=\sqrt{\frac{7\times 70}{4}}=3.5\sqrt{10}c{m}^2$

Problem 2: The outer radius of a hollow spherical shell is 14 cm, and the inner radius is 7 cm. Calculate the surface area of the outer surface and the inner surface of the shell.

Solution: Given that, 

Outer radius (R) = 14cm 

Inner radius (r) = 7cm 

The surface area of the outer surface = $4\pi R^2=4\left(\frac{22}{7}\right)14^2$

$=4\times 22\times 28=2464c{m}^2$

The surface area of the inner surface $4\pi r^2=4\left(\frac{22}{7}\right)7^2$

$=4\times 22\times 7=616c{m}^2$

Problem 3: The volume of a sphere is given as 904.32 cm³. Find the surface area of the sphere.

Solution: We know the Volume of the sphere (V) = $\frac{4}{3}\pi r^3$

$\frac{4}{3}\pi r^3=904.32$

$r^3=\frac{904.32\times 3}{4\pi }\approx 216c{m}^3$

$r=\sqrt[3]{216}=6cm$

Now, using Surface Area of the sphere(A) = $4\pi r^2$

$A=4\times \frac{22}{7}\times 6^2=\frac{3168}{7}$

$A=452.38c{m}^2$