mathschallenge.net

Problem

In the following diagram each of the squares have unit length sides.

Find the perimeter of the hexagon.

Solution

Consider the following diagram.

$\angle DAE = 360 - (60 + 90 + 90) = 120^o \implies \angle DAJ = \dfrac{120}{2} = 60^o$ and $\angle ADJ = 30^o$

$\therefore DJ = \cos(30^o) = \dfrac{\sqrt{3}}{2} \implies DE = \sqrt{3}$

Hence the perimeter of the hexagon is $3(\sqrt{3} + 1)$.

Find the area of the hexagon.