An n-sided polygon has four of the interior angles 154 degree, 151 degree, 149 degree and 140 degree and the rest of the exterior angles are 19.5 degree each. Find the value of n.

Question

An n-sided polygon has four of the interior angles 154 degree, 151 degree, 149 degree and 140 degree and the rest of the exterior angles are 19.5 degree each. Find the value of n.

Answer 1

There are two keys facts that help you solve this problem:

1. In any (convex) polygon, the exterior angle measures must add up to 360 total degrees.
2. Every interior angle has a "partner" exterior angle that is supplementary (makes 180 degrees) when added to the interior.

To get some visual intuition for this, here's a link to a picture of a pentagon (5-sided) with the sides extended to show all the exterior angles: http://www.geom.uiuc.edu/~dwiggins/pict09.GIF

So, the first step in your problem is to find all the exterior angles associated with the interior angles that they give you. These are 26 (180 - 154), 29 (180 - 151), 31 (180 - 149), and 40 (180 - 140).

Adding all of these up, we get 26+29+31+40 = 126. So, these four exterior angles account for 126 degrees out of the total 360 required of all exterior angles. This means we need to account for 234 (360 - 126) more degrees for the exterior angles.

Now the problem tells you that the remaining exterior angles all have the same angle measure, 19.5 degrees. Since we need to make up a total of 234 degrees, we need to know how many times 19.5 goes into 234. We have that 234/19.5 = 12. So now we know that, in addition to the four angles the problem told us about, there are 12 other angles in the polygon.

Putting this all together, we now have that there are a total of 16 (12 + 4) angles in the polygon. Since polygons must have an equal number of angles and sides, we know that the polygon must be 16-sided!