mathschallenge.net

Problem

Find the area of the annulus.

Solution

Surprisingly it is unnecessary to know the radius of either circle to determine the area of the annulus. Consider the following diagram, where $R$ is the radius of the large circle and $r$ is the area of the small circle.

As a tangent meets a radius at right angles, the triangle is right angled, so $R^2 = r^2 + d^2$.

So the area of the annulus, $\pi R^2 - \pi r^2 = \pi d^2 = 25\pi$.