# Solution to the Multiple Semi-Circles in a Circle Puzzle +-- {.image} [[MultipleSemiCirclesinaCircle.png:pic]] > The green area is double the yellow area. What fraction of the total is shaded? =-- ## Solution by [[Area of a Circle]] +-- {.image} [[MultipleSemiCirclesinaCircleLabelled.png:pic]] =-- As in the diagram, let $a$, $b$, and $c$ be the radii of the three smallest circles. The upper green circle has radius $a + b$, the lower green radius $b + c$, and the full circle has radius $a + b + c$. The area of the upper green region is: $$ \frac{1}{2} \pi (a + b)^2 - \frac{1}{2} \pi a^2 - \frac{1}{2} \pi b^2 = \frac{1}{2} \pi( a^2 + 2 a b + b^2 - a^2 - b^2 ) = \pi a b $$ Likewise, the area of the lower green region is $\pi b c$. The total green area is thus $ \pi b (a + c)$. This is double the yellow area, which is $\pi b^2$, so $a + c = 2 b$ and the total shaded area is $3 \pi b^2$. As $a + c = 2 b$, the radius of the outer circle is $3 b$ and so its area is $9 \pi b^2$. The fraction of this that is shaded is therefore $\frac{1}{3}$.