# Polygons with Equal Perimeter +-- {.image} [[PolygonswithEqualPerimeter.png:pic]] > These regular polygons have the same perimeter. Find the ratio of the area of the circles. =-- ## Solution by [[Lengths in a Regular Hexagon]] and [[Lengths in an Equilateral Triangle]] +-- {.image} [[PolygonswithEqualPerimeterLabelled.png:pic]] =-- As the triangle and hexagon have the same perimeter, the side length of the hexagon is half that of the triangle. Let $a$ be the side length of the hexagon. From [[lengths in a regular hexagon]], the height of the hexagon is $\sqrt{3} a$. The area of the circle is therefore $\frac{3}{4} \pi a^2$. From [[lengths in an equilateral triangle]], the radius of the [[incircle]] is $\frac{1}{3} \frac{\sqrt{3}}{2}$ times the side length of the triangle, so is $\frac{\sqrt{3}}{3} a$ so its area is $\frac{1}{3} \pi a^2$. The ratio of the area of the circles is therefore $4 : 9$.