Angle facts and angles in polygons
A few angle facts unlock most geometry questions. For polygons, the key ideas are that exterior angles always add up to 360°, and that the interior angles add up to (n − 2) × 180°.
Part 1 of 3: Learn it
In short
- Angles on a straight line add to 180°; around a point, 360°; in a triangle, 180°; in a quadrilateral, 360°.
- Interior angles of an n-sided polygon add to (n − 2) × 180°.
- Exterior angles of any polygon add to 360°; for a regular polygon each one is 360° ÷ n.
Where this is in your specification
Spec points: DfE G1 and G3
| Board | Topic: Angles, polygons and parallel lines |
|---|---|
| DfE content | G1, G3, G4, G6 |
| AQA 8300 | Geometry and measures |
| Edexcel 1MA1 | Geometry and measures |
| OCR J560 | 8 Basic geometry |
| Eduqas C300 | HG1, HG3, HG4, HG6 |
| Cambridge IGCSE 0580 | E4.1, E4.6 |
| National 5 C847 75 | Geometric skills: angles in shapes |
Basic angle facts
- On a straight line, the angles total 180°.
- Around a point, the angles total 360°.
- Where two straight lines cross, the angles facing each other (vertically opposite) match.
- Angles in a triangle add up to 180°, and in a quadrilateral to 360°.
- An isosceles triangle has two equal sides, and the two angles at the base of those sides are equal.
Interior and exterior angles
An interior angle is inside the polygon; the exterior angle is between one side and the next side extended. At each corner, interior + exterior = 180°.
The (n − 2) comes from splitting the polygon into triangles from one corner: a hexagon splits into 4 triangles, so its angles add to 720°.
Common polygons
| Polygon | Sides | Interior sum | Regular: each exterior | Regular: each interior |
|---|---|---|---|---|
| Pentagon | 5 | 540° | 72° | 108° |
| Hexagon | 6 | 720° | 60° | 120° |
| Octagon | 8 | 1080° | 45° | 135° |
| Decagon | 10 | 1440° | 36° | 144° |
What do the interior angles of an octagon add up to?
Show the answer
(8 − 2) × 180 = 1080°.
Part 2 of 3: See it worked
Worked examples
Example 1
A regular polygon has interior angles of 156°. Work out the number of sides.
- Each exterior angle is 180 − 156, which is 24°
- n = 360 ÷ 24 = 15
Answer: 15 sides.
Example 2
A hexagon has angles 100°, 130°, 125°, 110°, x and 2x. Find x.
- Interior sum: (6 − 2) × 180 = 720°
- 100 + 130 + 125 + 110 + 3x = 720
- 465 + 3x = 720, so 3x = 255
Answer: x = 85° (and 2x = 170°).
Common mistakes
- Using n × 180 instead of (n − 2) × 180 for the interior sum.
- Dividing 360 by the interior angle to find the number of sides. Use the exterior angle.
- Assuming a polygon is regular when the question does not say so.
- Giving numbers without reasons when the question asks for them.
Find each exterior angle of a regular 12-sided polygon.
Show the answer
360 ÷ 12 = 30°.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
What do the interior angles of an octagon add up to?
(8 − 2) × 180 = 1080°.
Find each exterior angle of a regular 12-sided polygon.
360 ÷ 12 = 30°.
Can a regular polygon have an interior angle of 130°?
No. The exterior angle would be 50°, and 360 ÷ 50 = 7.2 is not a whole number of sides.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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