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# How to Find the Sum of the Interior Angles of a Polygon

In the world of GMAT geometry, a large number of questions deal with polygons. A polygon is simply a geometric figure having three or more (usually straight) sides. In other words, a triangle is a polygon, and by far the largest percentage of polygon questions on the GMAT concern triangles. As such, be sure you’re up to speed on all of your triangle rules, especially the four most common triangles tested on the GMAT (see “GMAT Geometry – Triangles”).

Occasionally, you’ll see questions that deal with polygons of more than four sides. Often the key to answering these types of GMAT geometry questions lies in your ability to find the sum of the interior angles of the polygon in question. I’m going to teach you a really cool shortcut for doing that, but first, let’s be clear what we’re talking about when we say interior angle. Very simply put, an Interior Angle is an angle inside a geometric shape as illustrated in this diagram:

Interior and Exterior Angles of Polygons

In a triangle, as we all know, the sum of the measures of the interior angles sum to 180°.

But again, what about polygons of more than four sides? (Note: A polygon with four sides is called a quadrilateral, and its interior angles sum to 360°). Oftentimes, GMAT textbooks will teach you this formula for finding the sum of the interior angles of a polygon, where n is the number of sides of the polygon:

Sum of Interior Angles = (n – 2) * 180°

But as you know by now, I like to teach you how to get right answers without having to memorize a bunch of formulas whenever possible. This is an example of a theme tested on the GMAT where it’s not necessary for you to memorize the formula above. Instead, there’s a really cool shortcut that will enable you to save time on test day and easily find the sum of the interior angles of a polygon without using that annoying formula. Check this out:

For the sake of being thorough, I’ll include this table (below) of the most common polygons tested on the GMAT and the sum of their interior angles. One quick note about this table: A Regular Polygon is one where all of the sides of the polygon are the same length, and hence all of the interior angles will have the same degree measure as well. This isn’t a requirement of a polygon, certainly, but when it is the case (i.e. if the GMAT refers to a polygon as a regular polygon (don’t assume that, though)), you’ll know that all of the angles are equal and you seem them listed in the last column of the table.

Here’s one more quick but useful rule for you to know about polygons. We’ve just spent a lot of time looking at the sum of interior angles of a polygon, which differ based on the number of sides and can be quite large, but what about the sum of the exterior angles of a polygon? You’re going to like this:

Rule: The sum of the exterior angles of a polygon is 360°.

Always. No matter how many sides the polygon has. Pretty easy, huh?

Now that you’re an expert at finding the sum of the interior and exterior angles of a polygon, how might this concept be tested on the GMAT? Try your hand at this sample GMAT polygon question and type your answer in the “Comment” field below. I’ll be monitoring your responses and answering any questions you have, including a really easy way to think about this question (hint: no need for algebraic formulas). Good luck!

Question: The measure of the interior angles in a polygon are consecutive integers. The largest angle measures 110 degrees. How many sides does this polygon have?

(A) 5
(B) 6
(C) 7
(D) 9
(E) 11