# GMAT Math : Complex polygons

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

Is the sum of all the interior angles of a polygon less than 1440?

1. The polygon has 9, 10 or 11 sides
2. The polygon has 10, 11 or 12 sides Readers are encouraged to take some time to tackle the above question 🙂

Trick: The sum of the interior angles of ANY polygon with n sides is equal to (n – 2) x 1800

Statement (1) tells us that the polygon has 9, 10 or 11 sides – the implication on the interior angles is 12600, 14400 or 16200 – clearly not sufficient independently

Statement (2) tells us that the polygon has 10, 11 or 12 sides – the implication on the interior angles is 14400 or 16200 or 18000– clearly also not sufficient independently

Statement (1) and (2) tells us the interior angles can either be 14400 or 16200 – again not sufficient.

We go with option (E)

How do we know that the formula (n – 2) x 1800 always works – the reason is that any polygon can be decomposed into triangles as shown below. There are n triangles and each triangle’s internal angle sum measure 1800. If we subtract the center angle (around O) measuring 3600 we are left with 1800 x n – 360 = (n – 2) x 1800 