# Solution to the Swirling Semi-Circles Puzzle +-- {.image} [[SwirlingSemiCircles.png:pic]] > The largest and smallest semicircles are concentric. What’s the shaded area? =-- ## Solution by [[Circle Area]] and [[Pythagoras' Theorem]] +-- {.image} [[SwirlingSemiCirclesLabelled.png:pic]] =-- In the above diagram, $O$ is the centre of the smallest circle. As the largest and smallest semi-circles are concentric, the lengths of $O A$ and $O D$ are equal, so the lengths of $C A$ and $B D$ are also equal. These are the diameters of the two middle size semi-circles, so they are the same size. This means that the orange and green areas in the above diagram are [[congruent]]. The shaded area is therefore the same as the difference between the areas of the largest and smallest semi-circles. Let $a$ be the radius of the smallest semi-circle and $b$ of the largest. These two radii form two sides of the [[right-angled triangle]] $O B E$. So by [[Pythagoras' theorem]], $$ a^2 + 2^2 = b^2 $$ The area of the shaded region is therefore given by $\frac{1}{2} \pi b^2 - \frac{1}{2} \pi a^2 = \frac{1}{2} \pi \times 4 = 2 \pi$.