GCSE Maths Angles & Polygons Notes
Angle facts let you find missing angles using rules for straight lines, triangles, parallel lines and polygons. Always give a reason with each step.
Key concepts
- Angles on a line and at a point
- Angles on a straight line add to 180°; angles around a point add to 360°.
- Triangles and quadrilaterals
- Angles in a triangle sum to 180°; in a quadrilateral they sum to 360°.
- Parallel line angles
- Alternate angles are equal, corresponding angles are equal, co-interior angles add to 180°.
- Polygon angles
- Exterior angles of any polygon add to 360°; interior + exterior = 180° at each vertex.
Key formulas
Interior angle sum
sum = (n − 2) × 180°
Regular exterior angle
exterior = 360° ÷ n
Worked example
Find the size of each interior angle of a regular hexagon.
- 1A hexagon has n = 6 sides.
- 2Interior angle sum = (6 − 2) × 180° = 4 × 180° = 720°.
- 3Divide by 6 for a regular hexagon: 720° ÷ 6 = 120°.
Answer: 120°
Common mistakes
✗ Not giving a reason for each angle.
✓ Quote the rule, e.g. 'alternate angles are equal', to earn the reasoning mark.
✗ Confusing interior and exterior angles.
✓ They sit on a straight line together and add to 180°.
✗ Assuming a polygon is regular.
✓ Only divide equally if the shape is stated to be regular.
Quick check
Try these, then reveal the answers.
Angles in a triangle add to?Reveal answer
180°
Exterior angle of a regular pentagon?Reveal answer
72°
Interior angle sum of a quadrilateral?Reveal answer
360°
Exam tips
- Write the angle rule you use next to each step.
- Fill in every angle you can find, not just the one asked for.
- For regular polygons, the exterior angle method is often quickest.
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Frequently asked questions
What is the sum of interior angles of a polygon?+
(n − 2) × 180°, where n is the number of sides.
What do co-interior angles add up to?+
180° — they are the angles on the same side of a line crossing two parallel lines.
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