A regular polygon has all sides equal and all angles equal. For a regular polygon with n sides, the sum of interior angles is (n − 2) × 180°, each interior angle is (n − 2) × 180° ÷ n, and each exterior angle is 360° ÷ n. These three formulas solve every GCSE regular polygon angle question.
What is the difference between an interior and an exterior angle?
An interior angle is the angle inside the polygon at each vertex — the angle you would measure if you stood inside the shape looking outward.
An exterior angle is formed by extending one side of the polygon outward. At each vertex, the interior angle and exterior angle together make a straight line: interior + exterior = 180°.
For a regular polygon, every interior angle is identical and every exterior angle is identical, which makes the calculations clean and predictable.
What is the formula for the sum of interior angles?
For any polygon with n sides:
Sum of interior angles = (n − 2) × 180°
This formula comes from dividing the polygon into triangles by drawing diagonals from one vertex. An n-sided polygon splits into (n − 2) triangles, each contributing 180°.
| Polygon | n | Sum of interior angles |
|---|---|---|
| Triangle | 3 | (3 − 2) × 180° = 180° |
| Quadrilateral | 4 | (4 − 2) × 180° = 360° |
| Pentagon | 5 | (5 − 2) × 180° = 540° |
| Hexagon | 6 | (6 − 2) × 180° = 720° |
| Octagon | 8 | (8 − 2) × 180° = 1080° |
| Decagon | 10 | (10 − 2) × 180° = 1440° |
How do you find each interior angle of a regular polygon?
Because all angles are equal in a regular polygon, divide the sum by n:
Each interior angle = (n − 2) × 180° ÷ n
Worked example 1: regular hexagon (n = 6)
Each interior angle = (6 − 2) × 180° ÷ 6 = 4 × 180° ÷ 6 = 720° ÷ 6 = 120°
Worked example 2: regular octagon (n = 8)
Each interior angle = (8 − 2) × 180° ÷ 8 = 6 × 180° ÷ 8 = 1080° ÷ 8 = 135°
Worked example 3: regular 12-sided polygon (dodecagon)
Each interior angle = (12 − 2) × 180° ÷ 12 = 10 × 180° ÷ 12 = 1800° ÷ 12 = 150°
How do you find the exterior angle?
All exterior angles of any polygon sum to 360°. For a regular polygon:
Each exterior angle = 360° ÷ n
And since interior + exterior = 180°:
Interior angle = 180° − exterior angle
Worked example 4: regular pentagon (n = 5)
Exterior angle = 360° ÷ 5 = 72°
Interior angle = 180° − 72° = 108° (confirmed: (5−2)×180°÷5 = 540°÷5 = 108° ✓)
How do you find the number of sides when given an angle?
This is a common GCSE problem type — work backwards using the exterior angle formula.
Worked example 5: a regular polygon has each exterior angle = 24°. How many sides?
n = 360° ÷ 24° = 15 sides
Worked example 6: a regular polygon has each interior angle = 156°. How many sides?
Exterior angle = 180° − 156° = 24°.
n = 360° ÷ 24° = 15 sides
Either route gives the same answer. When given the interior angle, always convert to exterior angle first — the exterior angle formula is simpler to invert.
What if the number of sides doesn't divide evenly?
A regular polygon with integer sides must have an exterior angle that is a whole-number factor of 360°. If a calculation gives a non-integer n, re-check: either the polygon is not regular, or the question has a different structure. GCSE questions always produce integer solutions for n.
Frequently asked questions
Why does the exterior angle formula always give 360°?
Imagine walking around the perimeter of a polygon in one direction. At each vertex you turn through the exterior angle. By the time you return to your starting point, you have completed exactly one full turn — 360°. This is true for any convex polygon, regular or not.
Can a regular polygon have an interior angle greater than 180°?
No. A regular polygon is always convex, meaning all interior angles are less than 180°. As n increases, the interior angle approaches (but never reaches) 180°. An interior angle of exactly 180° would mean three points are collinear — not a vertex at all.
How do I remember which formula to use?
Use exterior angles when you need to find the number of sides (360 ÷ exterior = n). Use the interior angle formula when you are calculating an angle directly. Remember: the two angles at each vertex always add to 180°, so you can convert freely between them.
Do I need to memorise these formulas for GCSE?
The formula for the sum of interior angles — (n − 2) × 180° — is not always on the formula sheet, so it is worth memorising. The exterior angle result (360° ÷ n and the sum of 360°) is easy to derive from the walking argument. The more you practise applying them, the more natural they become.
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