Question

A square whose side is $2$meters has its corners cut away so as to form an octagon with all sides equal. Then, the length of each side of the octagon in meters is:

• A. $\frac{\sqrt{2}}{\sqrt{2}+1}$
• B. $\frac{2}{\sqrt{2}+1}$
• C. $\frac{2}{\sqrt{2}-1}$
• D. $\frac{\sqrt{2}}{\sqrt{2}-1}$

Answer 1

Let $ABCD$ be a square of side $2m$. We form a regular octagon by cutting all four corners.

Let $EF=FG=GH=HI=IJ=JK=KL=LE$$=x$

By symmetry, $AE=AL=BG=BF=CH=CI=DJ=DK=a$

Using Pythagoras theorem in $△ALE$, we get

${LE}^2={AL}^2+{AE}^2$

$x^2=a^2+a^2=2a^2\Rightarrow x=\sqrt{2}a$

And $AB=AE+EF+FB\Rightarrow 2=a+x+a\to 2a+x=2$

Put the value of x, we get

$2a+\sqrt{2}a=2\Rightarrow a\sqrt{2}(\sqrt{2}+1)=2\Rightarrow a=\frac{\sqrt{2}}{\sqrt{2}+1}$

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

$\therefore$Length of each side of the octagon is $\frac{2}{\sqrt{2}+1}m$