Geometry

Interior and exterior angles of a polygon: finding n from an angle

By Mr Gan · O-Level E Maths · Updated August 2026 · 7 min read

Why this topic trips students up

Polygon angle questions have two separate formulas living side by side, and the most common way to lose marks is reaching for the wrong one. Students often try to use 360° ÷ n for an interior angle (that formula gives the exterior angle) or plug a single angle straight into (n − 2) × 180° without realising that formula gives the sum of all interior angles, not one of them.

The fix is to keep the two facts separate in your head: the exterior-angle sum is always 360°, full stop, for any polygon, regular or not. The interior-angle sum depends on n and is only useful once you already know how many sides the shape has.

The method

For a regular polygon, the exterior-angle route is almost always faster than the interior-angle route, because 360 ÷ n is one step, while the interior version needs you to first find the sum with (n − 2) × 180° and then divide by n. Both give the same final answer, but the exterior angle shortcut saves a step and an opportunity to slip up.

Worked example 1: finding n from an interior angle

A regular polygon has interior angle 156°. Find the number of sides, n.

Worked example 2: finding a missing interior angle

An irregular hexagon (6 sides) has five known interior angles: 130°, 105°, 140°, 95° and 125°. Find the sixth angle.

Angle sum by shape

It helps to have a few interior angle sums memorised so you can sanity-check an answer quickly, without recalculating (n − 2) × 180° from scratch every time.

Every one of these shapes, regular or not, still has an exterior angle sum of exactly 360°. That fact does not change with the number of sides, which is exactly why it is the faster route into most regular-polygon questions. It also connects directly to angle facts on parallel lines: many exam questions combine polygon angles with alternate, corresponding and co-interior angles on parallel lines in the same diagram, so being fluent in both saves time when they appear together. The quadrilateral row above is also where the regular property tables for triangles and quadrilaterals and the circle-specific case of cyclic quadrilaterals pick up, since both build on the same interior angle sum.

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Frequently asked questions

Does the exterior angle sum of 360° apply to irregular polygons too?

Yes. Every simple convex polygon, regular or irregular, has exterior angles that add up to exactly 360°. The shortcut of dividing 360° by n only works for regular polygons, since it assumes every exterior angle is equal. For an irregular polygon you cannot find one exterior angle from n alone.

How do I know whether to use the interior or exterior angle formula first?

If the question is about a regular polygon and gives you one angle (interior or exterior), convert to the exterior angle if needed, then divide 360° by it to get n. If the question is about an irregular polygon and gives you several angles with one missing, use the interior angle sum (n − 2) × 180° and subtract the known angles.

Can n come out as a decimal or a fraction?

No. The number of sides of a polygon must be a whole number, so if your calculation gives something like n = 14.4, check your working: either the angle given in the question was rounded, or you used the interior angle where the exterior angle was needed.

What is a related topic worth checking next?

Congruence tests are a natural next step once polygon angle facts are solid, since many exam questions ask you to prove two triangles inside a polygon are congruent using SSS, SAS, AAS or RHS before using the angle facts to finish the question.

— Mr Gan Math Tuition