Notes
polygons with equal perimeter solution

Polygons with Equal Perimeter

Solution by Lengths in a Regular Hexagon and Lengths in an Equilateral Triangle

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 $4 : 9$.