Polygon Dilemma!

The sum of interior angle of a polygon is given by $s=(n-2)(180)$ where $n$ is the number of sides. We substitute $2014$ into the formula, we have

$2014=(n-2)(180)$

$\dfrac{2014}{180}+2=n$

$n \approx 13.19$

Therefore, the desired answer is $\boxed{13}$ .

Suppose that the polygon in question has n sides. Then we can say that the sum of all of its interior angles(let's denote it by S) is

S = 180(n - 2) degrees Since S does not exceed 2014, we have the inequality

180(n - 2) <= 2014 180n - 360 <= 2014 180n <= 2374 and n <= 13.1888888..... The maximum integral solution to this inequality is 13

180(n-2) : where n =no of sides 2014= 180(n-2) ..... solve it

For Any Polygon No Of Sides= ( summation of all it's internal angles/180)+2 So,For Our Problem No of Sides = (2014/180)+2=13.1888 But As No Of Sides Should Be An Integer , We Have To Choose The Lowest Nearest Integer Which Is (13) and our polygon summation of internal angles will be ((13-2) 180)=11 180=1980 degrees