# Suppose a circle of radius r is inscribed in a hexagon. What is the *perimeter* of the hexagon? Thanks!

Let length of each side of the regular hexagon circumscribing the circle of radius $r$ be $a$.
$\frac{\frac{a}{2}}{r} = \cot {60}^{\circ}$
$\implies a = 2 r \cot {60}^{\circ} = 2 r \cdot \frac{1}{\sqrt{3}}$.
So perimeter of the hexagon will be $= 6 a = \frac{12}{\sqrt{3}} r = 4 \sqrt{3} r$