# Area of a Ring or a Circular Washer (Circle Inside another Circle)

There are many pages in this site explaining how to find the area of a circle. Now is the time to apply this information to solve word problems involving circles.

In this presentation we will explore how to find the area of a ring or a circular washer. A circular washer consists of two concentric circles or circle inside a circle. Consider the following example:

Find the area of a circular washer which have an inner radius of 3 cm and the outer radius of 5 cm, as shown in the image given below: "Area of a ring" Now, if we look at the given washer (ring), there are two concentric circles. A smaller circle (inner circle) with the radius r = 3 cm and the larger circle (outer circle) with radius R = 5 cm. We used lower case letter r to represent the radius of inner circle and Upper case letter R to represent the radius of outer circle. To find the area of a circular washer or the ring, we need to find the area between both the circles which is yellow part as shown in the image at left. We'll use the following steps to do the task: 1. Find the area of the outer circle. 2. Find the area of inner circle. 3. Find the difference between the areas found in above steps.

Area of smaller circle:

Given radius for the smaller (inner) circle r = 3 cm

Area of smaller circle = pi x radius x radius = 3.14 x 3 x 3 = 3.14 x 9 = 28.26 sq.cm

Hence the area of the inner circle is 28.26 square centimeters.

Area of larger circle:

Given radius for the larger (outer) circle is R = 5 cm

Area of the larger circle = Pi x radius x radius = 3.14 x 5 x 5 = 3.14 x 25 = 78.50 sq.cm

Hence the area of the outer (larger) circle is 78.50 square centimeters.

Area of the washer: Once you have calculated the area of both the circles (inner and outer) using given radii, the next step is to find the area of the washer by subtracting the above results. The area of the washer is equal to the area between both the circles.

Area of the washer = Area of outer circle - Area of the inner circle

= 78.50 - 28.26 = 50.24 sq.cm

Hence the area of the ring (circular washer) with outer radius of 5 cm and the inner radius of 3 cm found to be 50.24 sq. cm.

 Shorter Method to do the same: Alternately the whole process can be shortened by using the following steps and little bit of more algebra: Area of the washer = area of outer circle - area of the inner circle Where "r" and "R" are the radii of inner and outer circles respectively.  Now pull "pi" out as gcf and we solve our equation as shown below: Hence the area of the ring (washer) is 50.24 sq.centimeters, same as the previous method.

Worksheets on finding area of a ring or area of a circular washer worksheets: