Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who need interior and exterior angle rules for any polygon — especially regular polygons — to become reliable mark-scorers.
What query it owns: how to find interior and exterior angles of polygons in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the revision-guide angle, while Tutopiya’s Polygons subtopic page owns the learning resource and the free Polygons quiz owns the practice.

Polygons extend the angle theorems you already know into shapes with any number of sides. Cambridge IGCSE Mathematics (0580/0607) tests the sum of interior angles, the sum of exterior angles, and angle facts for regular polygons. This guide gives you the formulas, shows how to apply them in exam stems, and links to targeted practice.

Key takeaways

• Sum of interior angles of an n-sided polygon: (n − 2) × 180°.
• Sum of exterior angles of any convex polygon: 360° (one at each vertex).
• In a regular n-sided polygon, each interior angle = (n − 2) × 180° / n.
• Each exterior angle of a regular n-gon = 360° / n.

What are polygons in Cambridge IGCSE Maths?

A polygon is a closed shape with straight sides. Cambridge IGCSE Extended papers ask you to find the number of sides from angle information, calculate interior or exterior angles of regular polygons, and solve multi-step diagram problems combining polygon rules with Angle Theorems.

Read the full notes on Tutopiya’s Polygons subtopic page before you attempt questions.

The polygon formulas you must know

Quantity	Formula	Example: hexagon (n = 6)
Sum of interior angles	(n − 2) × 180°	(4) × 180° = 720°
Each interior angle (regular)	(n − 2) × 180° / n	720° / 6 = 120°
Sum of exterior angles	360° (always)	360°
Each exterior angle (regular)	360° / n	360° / 6 = 60°

Interior and exterior angles at each vertex are supplementary: they sum to 180°.

How to solve polygon angle problems — step by step

1. Identify n — the number of sides — or work it out from the question.
2. Choose the right formula — interior sum, exterior sum, or single-angle formula for regular polygons.
3. Substitute and solve for the unknown.
4. Check — interior + exterior at one vertex should equal 180° for a regular polygon.
5. State the answer with units (degrees).

Test yourself with the free Polygons quiz.

Regular vs irregular polygons

Type	Interior angles	Exterior angles
Regular	All equal	All equal
Irregular	Vary	Vary, but still sum to 360°

For irregular polygons you can still use (n − 2) × 180° for the total interior angle sum if you know n.

Polygons in past-paper wording: command words that matter

Command word / phrase	What the question wants	Typical stem
Find the sum of the interior angles	Apply (n − 2) × 180°	“Find the sum of the interior angles of a decagon.”
Calculate the size of each interior angle	Regular polygon single angle	”Calculate the size of each interior angle of a regular pentagon.”
Find the number of sides	Solve for n from angle data	”Each exterior angle of a regular polygon is 30°. Find n.”
Work out	Calculate with method	”Work out the size of angle x in the polygon.”
Show that	Prove a given result	”Show that the interior angle of a regular hexagon is 120°.”

Worked exam-style stems (how to answer the wording)

1. “Find the sum of the interior angles of a 15-sided polygon.” (15 − 2) × 180° = 13 × 180° = 2340°. Reward: correct formula and answer.

2. “Each interior angle of a regular polygon is 162°. Find the number of sides.” Exterior angle = 180° − 162° = 18°. n = 360° / 18° = 20 sides. Reward: exterior angle method or interior formula rearranged.

3. “Show that each exterior angle of a regular octagon is 45°.” n = 8 → exterior = 360° / 8 = 45°. “Show that” requires the calculation written out.

When you can recognise the wording instantly, work the full set on the Geometry topical past papers questions and the Polygons quiz.

How polygons connect to Geometry

Polygon angle sums build on Angle Theorems and prepare you for Symmetry and tessellation-style questions. Use the Cambridge IGCSE Maths resource hub to revise weak areas.

Common mistakes students make

• Using 360° × n instead of (n − 2) × 180° for interior angle sums.
• Forgetting exterior angles always sum to 360°, not 180°.
• Applying the regular polygon single-angle formula to irregular shapes.
• Confusing interior and exterior angles when finding n.

When you need more support

If polygon angle questions keep costing marks, work through the Angle Theorems quiz and the Geometry topical past papers, then get help from a Cambridge IGCSE Maths tutor.

Frequently asked questions

What is the sum of interior angles of a polygon with n sides? (n − 2) × 180°. A triangle (n = 3) gives 180°; a quadrilateral gives 360°.

Do exterior angles always add up to 360°? Yes, for any convex polygon — one exterior angle at each vertex, taken in the same direction.

How do I find the number of sides from an exterior angle? For a regular polygon, n = 360° ÷ (each exterior angle).

How should I revise polygons? Memorise both formulas, work five mixed questions including finding n, then take the Polygons quiz.

Ready to master Cambridge IGCSE Maths polygons?

Start with the Polygons subtopic page, then book a free trial with a Cambridge IGCSE Maths specialist to turn polygon questions into guaranteed marks.

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