# In a regular polygon each interior angle is 135° greater than each exterior angle. How many sides has the polygon?

Dec 21, 2017

The polygon is Hexakaidecagon having $16$ sides.

#### Explanation:

Exterior angle of regular polygon of $n$ sides is $E = \frac{360}{n}$

Interior angle of regular polygon of $n$ sides is $I = \frac{180 \left(n - 2\right)}{n}$

Given condition , $I = E + 135 \therefore \frac{180 \left(n - 2\right)}{n} = \frac{360}{n} + 135$ or

$\frac{180 \left(n - 2\right)}{n} - \frac{360}{n} = 135$. Multiplying by $n$ on both sides

we get , $180 \left(n - 2\right) - 360 = 135 n$ or

$180 n - 360 - 360 = 135 n \mathmr{and} 180 n - 135 n = 720$ or

$45 n = 720 \mathmr{and} n = \frac{720}{45} \mathmr{and} n = 16$

The polygon is Hexakaidecagon having $16$ sides. [Ans]