Question

A regular octagon is formed by cutting congruent isosceles right-angled triangles from the corners of a square. If the square has side-length $1$, the side-length of the octagon is$?$

• A. $\frac{\sqrt{2}-1}{2}$
• B. $\sqrt{2}-1$
• C. $\frac{\sqrt{5}-1}{4}$
• D. $\frac{\sqrt{5}-1}{3}$

Answer 1

The correct option is B $\sqrt{2}-1$

Let one of the congruent sides of the isosceles triangle be $x$

The hypotenuse of the triangle would be $\sqrt{2}x$

Since it is a regular octagon, all the sides are equal.

$\Rightarrow x+x+\sqrt{2}x=1$

$\therefore x=\frac{1}{2+\sqrt{2}}=\frac{1}{2+\sqrt{2}}\times \frac{2-\sqrt{2}}{2-\sqrt{2}}=\frac{2-\sqrt{2}}{2}$

The side length of the octagon is $\sqrt{2}x=\sqrt{2}\times \frac{2-\sqrt{2}}{2}=\sqrt{2}-1$