Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Polygons

In one sentence. Interior and exterior angles of any polygon — driven by the one unbreakable fact that exterior angles always total 360°.

Geometry — one of 4 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Both tiers.

This module ends with Johannes Kepler. Learn Polygons, teach it, then defend it against Johannes Kepler.

Where this sits in the specification

Specification 4MA1, 4.2 Polygons. Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 4.2 — Polygons

1 · Understand it

What it is

For any polygon: the EXTERIOR angles sum to 360°, always. The interior angles sum to (n − 2) × 180°. For regular polygons (all angles equal): exterior = 360/n, interior = 180 − exterior. Every polygon question is some rearrangement of these three lines.

Why it works

Walk around the boundary of any polygon: at each corner you turn by the exterior angle, and by the time you're home you've made exactly one full turn — 360°, whatever the shape. The interior sum follows because a polygon fans into (n − 2) triangles from one vertex. And interior + exterior = 180° because they sit together on a straight edge.

When a question is asking for it

'How many sides?' from an angle (go via the exterior angle — it divides 360), interior-angle sums, and tiling/meeting-point questions where angles at a point must total 360°.

How to hold it in your head

The exterior angle is the polygon's 'turning per corner'. A 15-sided regular polygon turns 360/15 = 24° at each corner. Given an interior angle, subtract from 180 to get the turn — then 360 ÷ turn counts the corners. The exterior angle is almost always the fastest door in.

2 · The mistakes that cost marks

  • Exterior angles summing to (n − 2) × 180. That's the INTERIOR sum; exteriors always make exactly 360°.
  • Interior angle of a regular polygon as 360/n. That's the exterior; interior = 180 − 360/n.
  • Dividing 360 by the interior angle to count sides. Divide 360 by the EXTERIOR angle.
  • Using regular-polygon formulas on an irregular polygon. Only the SUMS hold for irregular shapes.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

A regular polygon has an exterior angle of 24°. How many sides has it?

  1. Exterior angles of any polygon sum to 360°.
  2. Sides = 360 ÷ 24 = 15.

Answer: 15 sides

Find the interior angle of a regular decagon.

  1. Exterior: 360 ÷ 10 = 36°.
  2. Interior: 180 − 36 = 144°.

Answer: 144°

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — Work out the sum of the interior angles of a hexagon. (answer in °)
  2. Grade 7 — Work out the interior angle of a regular decagon (10 sides). (answer in °)
  3. Grade 8 — A regular polygon has an exterior angle of 24°. How many sides does it have?
  4. Grade 9 — A regular polygon has an interior angle of 162°. How many sides does it have?
  5. 9+ — A regular pentagon, a square and a third regular polygon meet at a point with no gaps and no overlaps. How many sides does the third polygon have?

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Polygons, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Polygons works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Polygons against Johannes Kepler — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 10 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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