How to find the angles of a regular polygon — examples
How do you find the angles of a regular polygon?
In any polygon — regular or not — the sum of the interior angles depends only on the number of sides. When the polygon is regular (all sides and all angles equal), you can share that sum equally to find the size of each angle. In this guide we build the formula from scratch and solve every common type of question together — including finding the number of sides from a given angle.
Worked examples, step by step
What are the sum of the interior angles and the interior angle of a regular hexagon (n = 6)?
- Substitute n = 6 into the sum formula: (6 − 2) × 180
- Calculate: 4 × 180 = 720 — that is the sum of the interior angles
- Because the hexagon is regular, all six angles are equal: 720 ÷ 6 = 120
- Answer: the angles add to 720 degrees, and each interior angle is 120 degrees
What are the interior angle and the exterior angle of a regular pentagon (n = 5)?
- Sum of the interior angles: (5 − 2) × 180 = 3 × 180 = 540
- One interior angle (regular pentagon, all angles equal): 540 ÷ 5 = 108
- Exterior angle by the direct formula: 360 ÷ 5 = 72
- Check: an interior angle and an exterior angle add to 180: 108 + 72 = 180 — it works
Each interior angle of a regular polygon is 144 degrees. How many sides does it have?
- The exterior angle makes 180 with the interior angle: 180 − 144 = 36
- The exterior angle is always 360 divided by n, so 36 = 360 ÷ n
- Solve for n: n = 360 ÷ 36 = 10
- Check: (10 − 2) × 180 ÷ 10 = 8 × 180 ÷ 10 = 144 — it works. Answer: 10 sides
What is the sum of the interior angles of a 7-sided polygon, even if it is not regular?
- The sum formula depends only on the number of sides; it does not matter whether the polygon is regular: (n − 2) × 180
- Substitute n = 7: (7 − 2) × 180 = 5 × 180
- Calculate: 5 × 180 = 900
- Answer: the interior angles add to 900 degrees in every 7-sided polygon — regular or not
Now try it yourself
Frequently asked questions
Why is the formula for the angle sum (n − 2) × 180?
Because any convex polygon with n sides can be split into triangles by drawing diagonals from one vertex — and you get exactly n − 2 triangles. The angles in each triangle add to 180 degrees, so all the interior angles of the polygon add to (n − 2) times 180.
What is the difference between a regular polygon and an irregular one?
In a regular polygon all the sides are the same length and all the angles are the same size, so you can share the total angle sum equally between the angles. In an irregular polygon the sum of the angles is still (n − 2) × 180, but the individual angles can be different from one another.
Why is the exterior angle of a regular polygon always 360 divided by n?
When you walk around all the sides of the polygon and return to your starting point, you have turned through exactly one full turn — 360 degrees. In a regular polygon every turn (exterior angle) is the same, so each one is 360 divided by n.
Can a convex polygon have an interior angle of 200 degrees?
No. In a convex polygon every interior angle is less than 180 degrees. An angle of 200 degrees is only possible in a concave polygon, which has at least one reflex interior angle greater than 180 degrees.
✍️ Written by the MathsUK teamChecked against the National Curriculum and GCSE specificationsLast updated: