Interior Angle of Polygon Formula: Exterior Angles, Sides and Diagonals

A polygon is a closed figure with straight sides. Almost every polygon question in SSC and railway papers uses just two facts, both true for any convex polygon with n sides.

• The interior angles add up to (n − 2) × 180°. Drawing diagonals from one vertex cuts the polygon into n − 2 triangles, and each triangle contributes 180°.
• The exterior angles add up to 360°, whatever the number of sides. Walking once round the boundary turns you through exactly one full turn.

At each vertex the interior and exterior angles lie on a straight line, so they add to 180°. That links the two facts and lets you move between them.

In a regular polygon all sides and all angles are equal, so each angle is the sum divided by n.

The exterior angle is the fastest route to n, because n = 360° ÷ exterior angle. If the question gives the interior angle, subtract it from 180° first.

Ratio and difference questions use the same 180° link. If the interior and exterior angles are in the ratio p : q, the exterior angle is q/(p + q) × 180°.

When the angles are not all equal, the interior-sum rule still holds, so a missing angle is the sum minus the angles you know. This type appears often with pentagons and hexagons, where the sums are 540° and 720°.

Some questions give the angles in a pattern, such as consecutive terms of an arithmetic progression. Write them as a, a + d, a + 2d and so on, add them, and set the total equal to (n − 2) × 180°.

Each vertex joins to every other vertex except itself and its two neighbours, giving n − 3 diagonals per vertex. Every diagonal is counted from both ends, so divide by 2.

• ✘ Using n × 180° for the interior sum. It is (n − 2) × 180°.
• ✘ Giving the interior angle when the question asks for the exterior one.
• ✘ Counting n(n − 1) ÷ 2 for diagonals. That includes the n sides.
• ✘ Accepting a number of sides that is not a whole number.

Angles also matter in clock angle problems and counting figures. For more geometry formulas, see the Quant cheat code.

Do the exterior angles add to 360° for polygons that are not regular?

Yes, for any convex polygon. Walking once around the boundary turns you through one full turn, 360°, whatever the side lengths or angles. For regular polygons each exterior angle is simply 360° divided by the number of sides.

Can a regular polygon have an interior angle of 100°?

No. Its exterior angle would be 80°, and 360 ÷ 80 = 4.5 is not a whole number. A regular polygon exists only when 360 divided by the exterior angle gives a whole number of sides of at least 3.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

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