Notes
two concentric circles solution

Solution to the Two Concentric Circles Puzzle

Solution by Pythagoras Theorem

Let us write $R$ for the radius of the outer circle, so on the diagram above we have $R = O A$, and $r$ for the radius of the inner circle, so $r = O C$. In the diagram, $E$ is the midpoint of the chord $B C$ and let us write $h$ for the distance $O E$. Note that $O E$ meets the line $A B C D$ at a right-angle.

Since $O E A$ is a right-angled triangle, Pythagoras' Theorem says that

Since $O E C$ is a right-angled triangle, Pythagoras’ Theorem also says that

Eliminating $h^2$ from these gives

We can directly calculate that, or we can use the difference of two squares to write it as $(12.5 + 2.5)(12.5 - 2.5) = 15 \times 10 = 150$.

The area we want to calculate is the area of the large circle minus the area of the smaller one. This is:

Solution by Intersecting Chords Theorem

In this diagram, the line $F G$ is chosen to be the diameter that passes through $C$.

The intersecting chords theorem says that in the above diagram then:

With $R$ and $r$ as above, then $F C = R - r$ and $C G = R + r$ so then

Hence $R^2 - r^2 = 150$ and so the orange area can be calculated as: