Average Angles

The formula for the interior angle of an n-sided polygon is

$\ 180 (n-2)$

since we are referring to regular polygons, if we divide the interior angle by the number of sides, we will obtain the angle of every side. So our new expression is

$\ (180(n-2))/n$

Now, we are looking for the integers whose average interior angle is less than or equal to 170, so we create the equation

$\ ((180(n-2))/n) \leq 170$ for all $\ n \in \Bbb Z$ (all integer n’s)

now, we just solve this equation by simplifying:

$\ 180(n-2)<=170n$

$\ 180n-360<=170n$

$\ 10n<=360$

$\ n<=36$

This tells us that the greatest value that n could be is 36 . Therefore n can be in the integer range from 3-36 because there doesn’t exist a polygon with less than 3 sides.

Next we need to find the sum of all these possible n ’s. To do this, we can quickly find the sum from 1 to 36 and then subtract 3 (1 and 2), by using the formula:

$\ ((n(n+1)) /2)-3$

where n is our greatest number. So we then get:

$\ ((36*37)/2)-3$ or $\ (18*37)-3$ or $\ 666-3$ which equals

$\ \fbox{663}$