Area of a Circular Path or Area of a Ring

A circular ring is formed by two concentric circles (circles with the same center but different radii).

Let the radius of the larger circle R and the radius of the smaller circle r.

The width of the ring is, (R - r)(R + r)

The area of a circular ring is the between the areas of the two circles.

Also, Area of a circle = πr22πr2

Area of larger circle = πR2πr2

Area of smaller circle = πr2πR2

Area of ring = πR2 - πr2 = πR2−r2 = π(R + r)(R - r)

If you know the average radius 'm' (middle radius) and width 'w' of the ring:

R = m +

r = m -

Substituting these into our formula:

Area = π(m+w22 - m−w22) = π(m2 + mw + w24 - m2 + mw - w24) = 2πmwπmw2

This shows that the area is equal to the average circumference (2πm) multiplied by the width (w)