Area of the Circular Ring – with Examples and FAQs

Area of a circular ring

Two concentric circles of different diameters form a circular ring (an annulus), which is a two-dimensional figure. For example, a washer’s top and bottom surfaces and the cross-section of a concrete pipe are examples of annuli. The annulus is the region between the two circles.

 It is the coloured portion as shown in the figure:-

What is the area of the ring?

We know the area of a circle is given by π(x)², where ‘x’ is the radius of the given circle. 

Radii R and r specify the dimensions of the outer and inner circles respectively. To find the area of a circular ring, we can subtract the area of the inner circle from that of the outer circle. 

For calculating the area of an annulus, use the following formula.

Area of the circular ring (Annulus) = A = π(R²﹣r²)

Derivation of the area of the ring

It is possible to find the area of the circular ring by measuring the area of the outer circle and the inner circle. To get the answer, we need to subtract the area of the inner circle from that of the outer circle

We’ll use “R” for the outer circle’s radius and “r” for the inner circle’s radius.

Therefore,

Outer circle area =\pi R^{2}

Inner circle area =\pi r^{2}

In other words, the area of the annulus is the difference between the areas of outer and inner circles.

Hence, Its area is equivalent to:

A r=A-a=\pi R^{2}-\pi r^{2}=\pi\left(R^{2}-r^{2}\right)=\pi(R-r)(R+r)

Where, 

Ar = area of the circular ring,

A = area of the outer circle,

a  = area of the inner circle. 

Examples

1. A 14 cm broad path surrounds a circular lawn with a 360 cm diameter. Find the area of the path.(Use π value of 22/7)

Given:

The radius of the inner circle (r) = 180 cm

Radius of outer circle (R) =(180 + 14) = 194 cm

∴ The area of path  = 𝜋(𝑅 – 𝑟)(𝑅 + 𝑟)

                                  = \frac{22}{7}\times (194 +180)(194-180)

                                 =16456 \mathrm{~cm}^{2}

Hence the area of the path is 16456 cm².

2. A circular path has an outer diameter of 728 m and an inner diameter of 700 m. Find the circumference and the area of the circular path. (Use π value of 22/7)

A circular path has an outside diameter of 728 m.

Therefore, its radius (R) = (728/2) = 364 m

The inner diameter = 700 m

Therefore, the inner radius (r) = (700/2) = 350 m.

Hence, the circumference / breadth of the circular path is equal to (R – r) = 364 – 350 = 14m. 

Circular path area (𝐴)

= 𝜋(R + r)(R – r)

= 22/7(364 + 350) (364 – 350)

= (22/7) x 714 x 14

= ( 22 x 714 x 2 )

This is equivalent to 31,416 m².

In other words, the circular walkway has a surface area of 31416 m ².