# Geometry: Regular Polygon Properties

This aptitude practice question is a geometry question. Concept Tested: Sum of interior and sum of exterior angles of a regular polygon.

## Question

Each interior angle of a regular polygon is 120 degrees greater than each exterior angle. How many sides are there in the polygon?

1. 6 sides
2. 8 sides
3. 12 sides
4. 13 sides

Correct Answer      Choice (3). The polygion has 12 sides

## Explanatory Answer

Definition 1: All sides AND all angles of a regular polygon are equal.
Definition 2: A convex polygon is one in which the measure of all interior angles is less than 180o.
Key formula 1: Sum of the interior angles of a convex polygon with n sides = (n - 2) * 180
Key formula 2: Sum of the exterior angles of a convex polygon = 360o.

Let each exterior angle be Ao. Then each interior angle will be 120 + Ao.
The sum of an exterior and its corresponding interior angle is 180o.
Therefore, A + (120 + A) = 180 => A = 30o.

From key formula 2, sum of the exterior angles of a convex polygon = 360o.
i.e., if the polygon has 'n' sides, then n $$times$ A = 360 Number of sides of the regular polygon, n = $\frac{360}{\text{measure of one exterior angle}}$ = $\frac{360}{30}$ = 12 sides. The correct answer is Choice$3). The polygon has 12 sides.

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