Problem 25, in the diagram below the diameter of the inner circle forms part of the diameter of the outer circle. The outer circle has a chord of length 16 that is parallel to its diameter and is also a tangent to the inner circle. What is the area of the shaded region? So let's give names to the radii of the two circles. First let the radius of the outer circle be capital R and the radius of the inner circle be lowercase r. Then we know that the area of the large circle is pi times capital R squared and the radius of the small white circle is pi times lowercase r squared. So the shaded region, which is the difference between the two circles, is pi times capital R squared minus lowercase r squared. Now we need to find capital R squared minus lowercase r squared, and in this position it's kind of hard to do so. However, it doesn't really matter that the two circles are tangent on the outside on both of their circumferences. We can move the inner white circle and as long as we don't change the size of either circle, the difference capital R squared minus lowercase r squared will be the same. If we move the inner white circle to the center, we're also keeping the tangent chord on the outside the same because we're just moving this white circle along this diameter that contains the diameters of both circles and still does. Now we know that because the white circle is in the middle, if we draw this tangent, it's going to break this chord into two equal pieces. So this length from the middle to the edge of the circle is going to be 8. And because the chord is tangent to the circle, we have a 90 degree angle right here. So using Pythagoras's theorem, we get that capital R squared is equal to lowercase r squared plus 8 squared. And the difference capital R squared minus lowercase r squared is 8 squared. Now again, the shaded region is pi times this difference. So the shaded region is pi times 8 squared, which is 64 pi. And the answer is C.

concentric circles

chord length

Pythagorean theorem

shaded region

area calculation