In this chapter we will learn a critical technique relating to GMAT Polygon questions

Lets start with an example from GMAT Data sufficiency:

**Is the sum of all the interior angles of a polygon less than 1440 ^{0Β }?**

**The polygon has 9, 10 or 11 sides****The polygon has 10, 11 or 12 sides**

Readers are encouraged to take some time to tackle the above question π

β¦

β¦

β¦

β¦

**Trick: The sum of the interior angles of ANY polygon with n sides is equal to (n β 2) x 180 ^{0}**

Statement (1) tells us that the polygon has 9, 10 or 11 sides β the implication on the interior angles is 1260^{0}, 1440^{0} or 1620^{0} β clearly not sufficient independently

Statement (2) tells us that the polygon has 10, 11 or 12 sides β the implication on the interior angles is 1440^{0} or 1620^{0} or 1800^{0}β clearly also not sufficient independently

Statement (1) and (2) tells us the interior angles can either be 1440^{0} or 1620^{0} β again not sufficient.

We go with **option (E)**

How do we know that the formula (n β 2) x 180^{0} always works β the reason is that any polygon can be decomposed into triangles as shown below. There are n triangles and each triangleβs internal angle sum measure 180^{0}. If we subtract the center angle (around O) measuring 360^{0} we are left with 180^{0} x n β 360 = (n – 2) x 180^{0}