Question

Each interior angle of a polygon is ${135}^{\circ }$. How many sides does it have?

$\left(a\right)8$

$\left(b\right)7$

$\left(c\right)6$

$\left(d\right)10$

Answer 1

The correct option is (a): $8$

Each interior angle of a polygon $={135}^{\circ }$

Sum of all the interior sides of a polygon $=(n-2)\times {180}^{\circ }$

$=(n-2)\times 2\times {90}^{\circ }=(2n-4)\times {90}^{\circ }$

Each side is given by $\frac{(2n-4)\times {90}^{\circ }}{n}$

$\therefore \frac{2n-4}{n}\times {90}^{\circ }={135}^{\circ }$

$\Rightarrow \frac{2n-4}{n}=\frac{135}{90}$

$\Rightarrow \frac{2n-4}{n}=\frac{3}{2}$

$\Rightarrow 4n-8=3n$

$\Rightarrow 4n-3n=8$

$\Rightarrow n=8$ sides