# Three Semi-Circles +-- {.image} [[ThreeSemiCircles.png:pic]] > The three coloured stripes are each the same width. If the smallest semicircle has area $5$, what's the area of the largest? =-- ## Solution by [[Pythagoras' Theorem]] and [[Circle Area]] +-- {.image} [[ThreeSemiCirclesLabelled.png:pic]] =-- Let $a$, $b$, $c$ be the radii of the three semi-circles in increasing order. With the points labelled as above, then $a$ is the length of $B F$, $b$ of $A D$ and $D E$, and $c$ of $C E$. Each of $A B$, $B C$, and $C D$ are one third of the radius of the middle semi-circle, so are $\frac{b}{3}$. Applying [[Pythagoras' theorem]] to triangle $F B D$ gives the relationship: $$ b^2 = a^2 + \left(\frac{2 b}{3} \right)^2 $$ so $\frac{5}{9} b^2 = a^2$. Applying [[Pythagoras' theorem]] to triangle $C D E$ gives the relationship: $$ c^2 = b^2 + \left( \frac{b}{3} \right)^2 $$ so $c^2 = \frac{10}{9} b^2 = 2 a^2$. The area of the larger semi-circle is $\frac{1}{2} \pi c^2 = \pi a^2$. The area of the smallest semi-circle is $\frac{1}{2} \pi a^2$, which is $5$, so the largest semi-circle has area $10$.