# Nested Circles and Polygons +-- {.image} [[NestedCirclesandPolygons.png:pic]] > Two circles, a rectangle and a regular hexagon, all neatly packed inside each other. What fraction of the outer circle is shaded? =-- ## Solution by [[hexagon|Properties of a Regular Hexagon]] and [[Pythagoras' Theorem]] +-- {.image} [[NestedCirclesandPolygonsLabelled.png:pic]] =-- Let $r$ be the radius of the inner circle and $R$ of the outer. The side length of the hexagon is then $\frac{2}{\sqrt{3}} r$ and these lengths are the sides of a [[right-angled triangle]], so by [[Pythagoras' theorem]] $$ R^2 = r^2 + \left(\frac{2}{\sqrt{3}} r\right)^2 = r^2 + \frac{4}{3} r^2 = \frac{7}{3} r^2 $$ Therefore the shaded region has area $\frac{3}{7}$ths of the outer circle.