# Minimum Number of Sides of a Polygon

Question

What is the minimum number of sides a polygon can have if all its interior angles are either 167 or 168 degrees only?

Solution

Let m and k be the number of angles with values 168 & 167 respectively.

Then the

sum of the interior angles will be

168m + 167k

Counting this in a different way:

But the polygon has (m + k) sides.

So this sum of the interior angles

(n – 2) x 180

Where n = number of sides of a polygon.

In our case,

n = m + k

So, we have to solve (or find the positive integer solutions to the equation):

168m + 167k = (m + k – 2) x 180

Which is obtained by combining the two previous facts.

And among the possible integer solutions choose the one with the “smallest m + k” as the answer.

The rest of the solution is shown in the picture: