Circles: Finding the Area of a Ring

A ring is made up of two circles same center point. Such circles are called concentric.

Problem

A circular hot tub with radius 6 feet has a sitting area around its perimeter, with a width of 1 foot. Find the area of the sitting area.

Strategy

A circle is completely defined by its radius, and its area is given by  A _circle =π*r². The tub and the sitting area form two concentric circles.

We know the radius of the outer circle (the tub) – so we can find its area. We know the width of the sitting area, from which we can calculate the inner radius (from the center of the tub to the inner edge of the sitting area), and find the area of the inner circle.

Subtracting the two areas gives us the area of the sitting area.

Solution

(1) R_tub= 6
(2) A_outer =π*R²=π*6²=36π
(3) r_inner= 6-1=5
(4) A_inner =π*r²=π*5²=25π
(5) A_ring =A_outer-A_inner=36π-25π=11π≈34.56