# Annulus

An annulus is a flat, ring-shaped structure made out of 2 circles. Another way of looking at it is a circular ring with a circular hole in it. Mathematically, it is the area between the 2 circles having a common center (concentric circles) lying in the same plane. Annulus is expressed in square units such as m2, cm2, in2, and ft2.

The term ‘annulus’ (plural – annuli) is derived from the Latin word, meaning ‘little ring’. Some real-life example of annulus are doughnut and finger ring.

Shown below is an annulus where a smaller circle lies inside a bigger one with ‘O’ as there center.

## Formulas

### Area

The area of an annulus can be obtained by finding the area of the outer circle and the area of the inner circle and then subtracting the area of the inner circle from the outer one.

The formula to calculate the area of an annulus is given below:

Derivation

Let,

Area of the outer circle = πR2, here R = radius of the outer circle

Area of the inner circle = πr2, here r = radius of the inner circle

∴ Area (A) of an annulus = πR2 – πr2

= π(R2 – r2)

The above equation can also be written in the form

Area (A) = π(R+r)(R-r)

Thus, the area of an annulus is given by the equation (A) = π(R2 – r2) or π(R+r)(R-r)

Let us solve some examples to understand the concept better.

Find the area of an annulus with an outer radius 16 cm and an inner radius 8 cm?

Solution:

As we know,
Area (A) = π(R2 – r2), here π  = 3.141, R = 16 cm, r = 8 cm
= 3.141(162 – 82)
= 3.141(256 – 64)
= 3.141 × 192
= 603.072 cm2

If the area of an annulus is 883 inches and its width is 2 inches, then calculate the radii of the inner and outer circles. (Use π = 3.141)

Solution:

As we know, the inner radius be r and the outer radius be R
According to the problem,
Width (w) = R – r, here w = 2 inches
=> 2 = R – r
=> R = 2 + r
Now,
Area (A) = π(R2 – r2)
=> 883 = π(R + r)(R – r)
=> 883 = 3.141(2 + r + r)(2)
=> 2 +2r = 883/3.141 × 2
=> 2 +2r = 883/6.282
=> 2 +2r = 140.56
=> 2r =140.56 – 2
=> 2r = 138.56
=> r = 69.28 inches
Thus, the radius (r) of the inner circle is 69.28 inches
Now, the radius (R) of the outer circle = (2 + 69.28) inches
= 71.28 inches

### Perimeter

The perimeter of an annulus is the total distance covered around the boundary of the outer and the inner circle. The formula to calculate the area of an annulus is given below:

Perimeter (P) = (R + r), here π = 3.141 = 22/7, R = radius of the outer circle, r = radius of the inner circle

Derivation

Let,

Perimeter (P) of the outer circle = 2πR, here R = radius of the outer circle

Perimeter (P) of the inner circle = 2πr, here r = radius of the inner circle

∴ Perimeter (P) of the annulus = 2πR + 2πr

= 2π(R + r)

Thus, the perimeter of an annulus is given by the equation (P) = 2π(R + r)

Let us solve an example to understand the concept better.

Find the perimeter of an annulus with outer radius of 15 cm and inner radius of 10 cm.

Solution:

As we know,
Perimeter (P) = 2π(R + r), π = 3.141, here R = 15 cm, r = 10 cm
= 2 × 3.141(15 + 10)
= 2 × 3.141 × 25
= 157.05 cm