 # Annulus

An annulus is a shape made out of two circles. It is a plane figure formed by two concentric circles. The region covered between two concentric circles is called annulus. It has a ring shape and has many applications in Mathematics. Some of the real-life examples are finger-ring, dough-nut, etc. The area of the annulus is determined if we know the area of circles (both inner and outer). The formula to find annulus of the circle is given by:

A = π(R2-r2)

where ‘R’ is the radius of outer circle and ‘r’ is the radius of inner circle. Here, we will learn its complete meaning and its area along with examples.

The circle is a fundamental concept not only in Maths but also in many fields. By its definition, we know, a circle is a plane figure that is made up of the points situated at the same distance from a particular point. See a figure of a circle given here. It shows a complete circle with some radius.  Now if the same circle is surrounded by another circle with some space in between them and radius bigger than this circle, the region formed in between the two circles is basically the annulus. Let us learn its meaning in terms of geometry along with the area formula and solved examples based on it.

## Annulus Meaning

The word “annulus” (plural – annuli) is derived from the Latin word, which means “little ring“. An annulus is called the area between two concentric circles (circles whose centre coincide) lying in the same plane.

It is the region bounded between two circles which share the same centre. This shape resembles a flat ring. It can also be considered as a circular disk having a circular hole in the middle. See the figure here showing an annulus. Here, two circles can be seen, where a small circle lies inside the bigger one. The point O is the centre of both circles. The shaded coloured area, between the boundary of these two circles, is known as an annulus. The smaller circle is known as the inner circle, while the bigger circle is termed as the outer circle.

In other words, any two-dimensional flat ring-shaped object that is formed by two concentric circles is called an annulus.

## Area of Annulus

The area of the annulus can be calculated by finding the area of the outer circle and the inner circle. Then we have to subtract the areas of both the circles to get the result. Let us consider a figure: In the above figure, two circles are having common centre O. Let the radius of outer circle be “R” and the radius of inner circle be “r”. The shaded portion indicates an annulus. To find the area of this annulus, we are required to find the areas of the circles.

Therefore,

Area of Outer Circle = πR2

Area of Inner Circle = πr2

Area of Annulus = Area of Outer Circle – Area of Inner Circle

Hence,

 Area of Annulus = π(R2-r2)

Or we can also write it as;

 Area of Annulus = π(R+r)(R-r)

### Annulus Examples

Q.1: Calculate the area of an annulus whose outer radius is 14 cm and inner radius 7 cm?

Solution: Given that outer radius R = 14 cm and inner radius r = 7 cm

Area of outer circle = πR2 = 22/7 x 14 x 14

= 22 x 14 x 2

= 616 cm2

Area of inner circle = πr2 = 22/7 x 7 x 7

= 22 x 7

= 154 cm2

Area of the annulus = Area of the outer circle – Area of the inner circle

Area of the annulus = 616 – 154

Area of the annulus = 462 cm2

Q.2: If the area of an annulus is 1092 inches and its width is 3 cm, then find the radii of the inner and outer circles.

Solution: Let the inner radius of an annulus be r and its outer radius be R.

Then width = R – r

3 = R – r

R = 3 + r

We know,

Area of the annulus = π(R2−r2)

or

Area of the annulus = π (R + r) (R – r)

1092 = 22/7 (3 + r + r) (3)

3 + 2r = 1092×722×3

3 + 2r = 115.82

2r = 115.82 – 3

2r = 112.82

r = 56.41

R = 3 + 56.41

= 59.41

So, Inner radius = 56.41 inches