# Area of a Ring – Formula and Examples

Here you will learn what is the formula for the area of circular ring and examples based on it.

Let’s begin –

## Area of a Ring

Let R and r be the outer and inner radii of a ring.

Then, area of a ring = $$\pi(R^2 – r^2)$$

#### Derivation :

We know that area of circle = $$\pi r^2$$

To calculate the area of ring (concentric circle), we subtract area of smaller circle from the area of bigger circle.

So, Area of rings = Area of bigger circle  – Area of inner circle

= $$\pi R^2$$ – $$\pi r^2$$

= $$\pi(R^2 – r^2)$$

Hence, the area is $$\pi(R^2 – r^2)$$.

Example : Find the area of the circular ring whose outer radius is 5 cm and inner radius is 4 cm.

Solution : Here, R = 5 cm and r = 3 cm

So, Area = $$\pi(R^2 – r^2)$$ = $$3.14 \times (25 -16)$$ = $$3.14 \times 9$$ = 28.26

Example : The area enclosed between two concentric is 770 $$cm^2$$. If radius of the outer circle is 21 cm, find the radius of the inner circle.

Solution : Radius of outer circle = 21 cm

Area enclosed between two concentric circles = 770 $$cm^2$$

Let the radius of inner circle be r.

Then, according to question

$$\pi (21)^2$$ – $$\pi r^2$$ = 770

$$\implies$$      $$21^2$$ – $$r^2$$ = $$770 \times 7\over 22$$

$$\implies$$  $$r^2$$ = 441 – $$70 \times 7\over 2$$ = 441 – 245 = 196

$$\implies$$   r = $$\sqrt{196}$$ = 14 cm

Hence, the radius of inner circle is 14 cm.