# 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 14400Β ?

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 π

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

Statement (1) tells us that the polygon has 9, 10 or 11 sides β the implication on the interior angles is 12600, 14400 or 16200 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 14400 or 16200 or 18000β clearly also not sufficient independently

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

We go with option (E)

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