SOLUTION: each exterior angle of a regular polygon is 45 degrees. what is the sum of the interior angles of the polygon?

Question 513928: each exterior angle of a regular polygon is 45 degrees. what is the sum of the interior angles of the polygon?
Answer by drcole(72) (Show Source): You can put this solution on YOUR website!
One of the theorems of Euclidean geometry is that the sum of the exterior angles of any polygon is 360 degrees. Since your problem specifies a regular polygon (one with equal length sides and angles), we can determine the number of angles (and therefore the number of sides) by dividing 360 degrees by 45 degrees.

So your regular polygon is eight-sided, i.e., an octagon.
The other important theorem we need to use here is that the interior and exterior angles are supplementary angles: in other words, their sum is 180 degrees. Since each exterior angle is 45 degrees, each interior angle is therefore 180 - 45 = 135 degrees. Imagine an octagon (like a stop sign) and see if this makes sense.
There are eight interior angles, each of 135 degrees. Therefore the sum of the interior angles is:
degrees
In general, the sum of the interior angles of a polygon with n sides is:
degrees
Thus the interior angles of a triangle sum to 180 degrees, the interior angles of a quadrilateral sum to 360 degrees, and so on. So you can get the same answer as above using the formula above with n = 8.
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