Question

In a convex polygon, sum of all the angles except one angle is ${2280}^o$. How many sides does the polygon have? What is the measure of this exceptional angle?

Answer 1

Given a convex polygon,

$\Rightarrow$ Sum of all angles except one angle $=2280°$

$\Rightarrow$ The sum of exterior angles $=360°$

If a convex polygon has $n$ sides, the sum of interior angles is ${180}^{o}n-360°$ or $(2n-4)$ right angles.

If the remaining angles are $x$ degrees

$\Rightarrow 180n-360=2280+x,$

$\Rightarrow 180n=2640+x$

and $n=\frac{2640+x}{180}.$

This must be an integer, $180|2640+x$ and $0<x<180.$

Putting $x=0,n=\frac{2640}{180}\approx 14.66$ and putting $x=180$ gives $\frac{2640+180}{180}\approx \mathrm{15.66.}$

The only integer in this range is $15.$ So the polygon has $15$ sides and remaining angle is $15\times {180}^o-{2640}^o=60°$

Hence, the answer is $60°.$