A unit square has its corner chopped off to form a regular polygon with eight sides. What is the area of the polygon? Source: ISI BMATH UGA 2017 paper
A regular polygon with 8 sides can be divided into eight congruent triangles .I tried to find the area of a triangle in the following method. An angle of a triangle is 360/8=45..then I draw a perpendicular bisector of the angle which is height of the triangle and I found it to be 1/2 as the square is of unit length .Now I got a right angled triangle from which I wanted to find the length of the base of the triangle but I couldn't do so.
Best Answer
If the side of the cut triangle is $x$, then in order to have a regular polygon, we need to have $$1-2x = \sqrt 2 x $$
solving for $x$ we get $$ x= 1- \frac {\sqrt 2 }{2}$$
The total area of the cut is $2x^2$ and the area of polygon is $$A=1-2x^2 = 2 (\sqrt 2 -1) \approx 0.8284$$