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Geometry & Measures · Foundation & Higher

Properties of polygons

A polygon is a closed 2D shape with straight sides. You need to know the properties of regular and irregular polygons, including angle sums and names.

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Revising properties of polygons the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Properties of polygons GCSE” on the site for the complete guide.

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Key facts to remember

  • 1Interior angle sum of a polygon with n sides = (n − 2) × 180°.
  • 2Each interior angle of a regular polygon = (n − 2) × 180° ÷ n.
  • 3Each exterior angle of a regular polygon = 360° ÷ n.
  • 4Interior + exterior angle = 180° (angles on a straight line).
  • 5The sum of all exterior angles of any polygon = 360°.
  • 6Regular polygons have all sides equal and all angles equal.
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Formulas

Interior angle sum
(n − 2) × 180°

n = number of sides

Each interior angle (regular)
(n − 2) × 180° ÷ n
Each exterior angle (regular)
360° ÷ n
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Worked examples

Example 1

Find the interior angle of a regular hexagon.

Working

  1. n = 6
  2. Interior angle sum = (6 − 2) × 180° = 4 × 180° = 720°
  3. Each interior angle = 720° ÷ 6 = 120°
Answer120°
Example 2

A regular polygon has exterior angles of 24°. How many sides does it have?

Working

  1. Number of sides = 360° ÷ exterior angle
  2. n = 360° ÷ 24° = 15
Answer15 sides
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Common mistakes

✗Using the interior angle sum formula for just one interior angle without dividing by n.
✗Confusing interior and exterior angles.
✗Forgetting that the exterior angle formula (360 ÷ n) only works for regular polygons.
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Exam tips

✓Learn the formula (n − 2) × 180° — it works for all polygons, regular or irregular.
✓For exterior angles: they always sum to 360° regardless of the number of sides.

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