Question Video: Finding the Area between Two Concentric Circles in Terms of 𝜋 Mathematics

Video Transcript

Find the area between two concentric circles of radii nine centimeters and 13 centimeters, respectively. Express your answer in terms of 𝜋.

Let’s begin by sketching a diagram of our two concentric circles. Concentric circles share the same center. So we will begin with a point 𝐶. That will be the center of both of our circles. The smaller circle we will sketch in pink, having a radius of nine centimeters. And then the larger circle, sharing the same center point, is sketched in orange with a radius of 13 centimeters.

Let’s recall the area 𝐴 of a circle is equal to 𝜋 times 𝑟 squared, where 𝑟 is the radius of the circle. We are looking for the area between the two circles. Let’s use blue to designate that area on our diagram. In order to find the area between the circles, it seems reasonable to begin by finding the area of each circle separately. First, we’ll find the area of the small pink circle by substituting nine for the radius. Since nine squared is 81, the area of our small circle is 81𝜋 centimeters squared.

The question asked us to give our answer in terms of 𝜋, which means that we don’t need to use an approximate decimal value for 𝜋 in our calculations. We can just leave it as a multiple of 𝜋. In fact, this is a great way to keep our answer as accurate as possible.

To find the area of the larger orange circle, we substitute 13 for the radius. And the result is 169𝜋 centimeters squared.

Now we must think about how to find the area between these two circles. If we start with the area of the larger orange circle and cut out the area of the smaller pink circle, that should leave us with the blue area between. So we will perform a subtraction. 169𝜋 centimeters squared minus 81𝜋 centimeters squared gives us 88𝜋 centimeters squared. Therefore, the area between these two concentric circles in terms of 𝜋 is 88𝜋 centimeters squared.