Polygons · Ages 13–14
Angle sum of a polygon
Angle sum = (n − 2) × 180°; a regular one's angle = sum ÷ n.
How to do it
Method
Angle sum = (n − 2) × 180°; a regular one's angle = sum ÷ n
Hexagon: (6 − 2) × 180° = 720°; regular, each 720° ÷ 6 = 120°
Another way
Cut from a point inside
6 × 180° − 360° = 720°
In full
The angles of an n-sided polygon add up to (n − 2) × 180° (reason: ∠ sum of polygon). Why? The diagonals from one vertex cut it into n − 2 triangles, each with angles adding to 180°. A hexagon: (6 − 2) × 180° = 720°. A regular polygon's angles are all equal, so each is the sum ÷ n: a regular hexagon's are 720° ÷ 6 = 120° each. For an unknown angle, work out the sum first, then take away the angles you know. Another way to cut it: join a point inside to every vertex, making n triangles, n × 180° in all, then take away the full turn of 360° in the middle: 6 × 180° − 360° = 720°, the same.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
The picture shows a pentagon. x =
Show the answer and working
Answer: 133°
A pentagon's angles add to (5 − 2) × 180° = 540° (∠ sum of polygon): x = 540° − 407° = 133°.
Example 2
The picture shows a hexagon. x =
Show the answer and working
Answer: 126°
A hexagon's angles add to (6 − 2) × 180° = 720° (∠ sum of polygon): x = 720° − 594° = 126°.
Example 3
What do the angles of a 6-sided polygon add up to?
- A. 720°
- B. 1080°
- C. 900°
- D. 360°
Choose the right answer
Show the answer and working
Answer: A. 720°
Angle sum = (n − 2) × 180° = (6 − 2) × 180° = 720° (∠ sum of polygon).
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The picture shows a pentagon. x =
Answer:
2.Each angle of a regular 8-sided polygon =
Answer:
3.What do the angles of a 9-sided polygon add up to?
- A. 1620°
- B. 1440°
- C. 360°
- D. 1260°
Choose the right answer
Answer:
4.The diagonals from A cut the polygon into triangles, so its angles add up to × 180° = °.
Fill in each blank
Answer:
5.In the pentagon, the three angles marked x are equal. x =
Answer:
All the answers
- 1. 131°
- 2. 135°
- 3. D. 1260°
- 4. 4, 4, 720
- 5. 110°
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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