# Solution to the Three Circles in a Semi-Circle Puzzle +-- {.image} [[ThreeCirclesinaSemiCircle.png:pic]] > What fraction of the semicircle is not covered by these three identical circles? =-- ## Solution by [[Lengths in a Regular Hexagon]] +-- {.image} [[ThreeCirclesinaSemiCircleLabelled.png:pic]] =-- By reflecting the diagram to create a full circle, it can be seen that the radius of the outer semi-circle is three times the radius of the smaller circle. This comes from the fact that joining the centres of the small circles makes a [[hexagon]]. Therefore the area of the semi-circle is $\frac{1}{2} \times 9$ times the area of one of the smaller circles. Since there are three such circles in the semi-circle, the area of the outer semi-circle is $\frac{3}{2}$ times the unshaded area, so the area of the shaded region is $\frac{1}{3}$ of the semi-circle.