Question

A piece of paper is in the shape of a square of side $1$m long. It is cut at the four corners to make a regular polygon of eight sides (octagon). The area of the polygon is

• A. $2(\sqrt{2}-1)m^2$
• B. $(\sqrt{2}-1)m^2$
• C. $\frac{1}{\sqrt{2}}{m}^2$
• D. none of these

Answer 1

The correct option is A $2(\sqrt{2}-1)m^2$

Clearly, $x+\sqrt{2}x+x=1$

So, $x=\frac{1}{2+\sqrt{2}}=\frac{2-\sqrt{2}}{2}$
Required area $=(1^2-4\times \frac{1}{2}{x}^2)m^2$
$=(1-2\times {\left(\frac{2-\sqrt{2}}{2}\right)}^2)m^2$
$=(1-\left(\frac{4+2-4\sqrt{2}}{2}\right))m^2$
$=(1-(3-2\sqrt{2}\left)\right)m^2$
$=2(\sqrt{2}-1)m^2$