A regular octagon is coloured as shown. What fraction of its area is red?

Solution

Exactly 1/2 of the figure is red.

First, notice that all sides have equal length. For convenience we scale it to 1. Now the distance from a side to the regular octagon centre is h and the total area is 8*h/2=4h.

The area of the middle red triangle is easily seen to be 2h/2=h. The others have area h₁/2 and h₂/2 respectively. Now since h₁+h₂=2h, they add up to also h.

Therefore the red area is 2h, which is half the total.