Hi Jobin Varghese,

That expression is the formula for the area of a regular octagon with side = a units.

You can prove it along the following lines:

Consider a regular hexagon ABCDEFGH, ABEF will form a rectangle and AHGF, BCDE will form two equal trapeziums whose sum of areas will be the area of an octagon.

We know that each interior angle of the octagon is 135 degrees.

=> AF = $$a+a\sin45+a\sin45=a\left(1+\sqrt{\ 2}\right)\ $$

Area of the rectangle = $$a\times\ a\left(1+\sqrt{\ 2}\right)\ =a^2\left(1+\sqrt{\ 2}\right)$$

Area of Trapezium = $$\dfrac{1}{2}\left(\dfrac{a}{\sqrt{\ 2}}\right)\left(\dfrac{\left(a+a\left(1+\sqrt{\ 2}\right)\right)}{2}\right)$$.

Total area of the octagon = $$2\times\ \left(\dfrac{1}{2}\left(\dfrac{a}{\sqrt{\ 2}}\right)\left(\dfrac{\left(a+a\left(1+\sqrt{\ 2}\right)\right)}{2}\right)\right)+a^2\left(1+\sqrt{\ 2}\right)$$ = $$2\times\ a^2\times\ \left(1+\sqrt{\ 2}\right)$$.

Regards.