Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / 5 / Solution

Let the equilateral triangle have side length $a$. Its altitude is $H=\sqrt{3}a/2$, and its incircle has radius $r=H/3=a/(2\sqrt{3})$: the centre lies on a median, one third of the altitude above the base. The area of the triangle is $aH/2=\sqrt{3}{a}^2/4$, while the circle has area $\pi r^2=\pi a^2/12$. ThereforeThe side length cancels, as required for a dimensionless ratio between similar configurations.