# Circles Around a Triangle +-- {.image} [[CirclesAroundaTriangle.png:pic]] > The diameters of these circles form a $3-4-5$ triangle. What's the total shaded area? =-- ## Solution by [[Dissection]] and [[Area of a Triangle]] +-- {.image} [[CirclesAroundaTriangleLabelled.png:pic]] =-- With the regions labelled as above, they combine as follows: * $A + B + C$ and $D + E + F + G + H$ are large semi-circles, write this as $L$ * $D + J$ and $B + E + F$ are medium semi-circles, write this as $M$ * $H + I$ and $C + G + F$ are small semi-circles, write this as $S$ * $E + F + G$ is the triangle, write this as $T$ Then the shaded region can be expressed as follows: $$ \begin{aligned} A + E + G + I + J &= A + B + C \\ &\phantom{=} + E + F + G \\ &\phantom{=} + H + I \\ &\phantom{=} + D + J \\ &\phantom{=} - B - C - F - H - D \\ &= L + T + S + M \\ &\phantom{=} - B - E - F \\ &\phantom{=} - C - G - F \\ &\phantom{=} - D - E - F - G - H \\ &\phantom{=} + E + F + G + E + F + G \\ &= L + T + S + M - M - S - L + T + T \\ &= 3 T \end{aligned} $$ As the triangle has side lengths $3$, $4$, and $5$ it is a [[right-angled triangle]] and has area $6$. So the shaded regions have total area $18$.