From a square of unit length,pieces from the corners are removed to form a regular octagon.Then, the value of area removed is?

Question

I'm having so much problem calculating the area, can anyone help?

Answer 1

Let $x$ be the length of the octagon. Then

$$x+x\cos45^\circ+x\cos45^\circ=1$$

So $\displaystyle x=\frac{1}{1+\sqrt{2}}=\sqrt{2}-1$.

The area removed is equal to the area of a square with side $x$.

The removed area is $x^2=3-2\sqrt{2}$.