# Solution to the Seven Circles Puzzle +-- {.image} [[SevenCircles.png:pic]] > The centres of all seven circles are marked. The width of the shaded ring is $1$. What's its area? =-- ## Solution by [[Pythagoras' Theorem]] +-- {.image} [[SevenCirclesLabelled.png:pic]] =-- Let $a$ be the radius of the blue circles (and therefore also of the inner black circle) and let $b$ be the radius of the pink. Then $O C$ has length $2 a$ and $O D$ has length $a + 1 + b$. As these are radii of the outer circle, they are equal and so $a = b + 1$. The length of $A B$ is $a + b = 2a - 1$ so by applying [[Pythagoras' theorem]] to triangle $O A B$, the following equation for $a$ is obtained: $$ \begin{aligned} a^2 + (a + 1)^2 &= (2 a - 1)^2 \\ 2 a^2 + 2 a + 1 &= 4 a^2 - 4 a + 1 \\ 2 a^2 - 6 a &= 0 \\ a &= 3 \end{aligned} $$ The area of the pink ring is then: $$ \pi (a + 1)^2 - \pi a^2 = (2 a + 1) \pi = 7 \pi $$