I was studying for some quizzes when I stumbled across this question. It goes like this:
Two regular quadrilaterals vinyl tiles each of $1$ foot long on each sides overlap each other such that the overlapping region is a regular octagon. What is the area of the overlapping region?
My work:
I imagined the problem like this:
The area of the shaded blue region is what I want to get. I don't know if one side of the octagon is one-third of that of the side of square, because my calculated area is slightly larger than that of my book's answer.
How do you get the area of the blue region above?
Best Answer
Let $a$ be a length-side of the octagon $ABCDE...$, where $BC$ placed on the square side.
Thus, since $\sin45^{\circ}=\frac{1}{\sqrt2}$, we obtain:$$\frac{1}{2}-\frac{a}{2}=a\cdot\frac{1}{\sqrt2},$$ which gives $a=\sqrt2-1$.
Thus, $\frac{1}{2}-\frac{a}{2}=\frac{1}{2}-\frac{\sqrt2-1}{2}=\frac{2-\sqrt2}{2}$ and the needed area is $$1-4\cdot\frac{1}{2}\left(\frac{2-\sqrt2}{2}\right)^2=1-\frac{1}{2}(2-\sqrt2)^2=2\sqrt2-2$$