A polygon is a closed, flat shape with three or more straight sides. Polygons are named by the number of sides — from triangle (3 sides) to dodecagon (12 sides) and beyond. At KS3, you must recognise each name, distinguish regular from irregular polygons, and apply the interior angle sum formula (n − 2) × 180°.
What are the names of the common polygons?
| Number of sides | Name | Interior angle sum |
|---|---|---|
| 3 | Triangle | 180° |
| 4 | Quadrilateral | 360° |
| 5 | Pentagon | 540° |
| 6 | Hexagon | 720° |
| 7 | Heptagon | 900° |
| 8 | Octagon | 1 080° |
| 9 | Nonagon | 1 260° |
| 10 | Decagon | 1 440° |
| 12 | Dodecagon | 1 800° |
The pattern: each time you add a side, the interior angle sum increases by 180°.
What is the interior angle sum formula?
For any polygon with n sides:
$$\text{Interior angle sum} = (n - 2) \times 180°$$
Why does this work? Drawing all the diagonals from one vertex divides the polygon into (n − 2) triangles, each with an angle sum of 180°. Multiplying gives the total.
Example: Find the interior angle sum of a heptagon (7 sides).
(7 − 2) × 180° = 5 × 180° = 900° ✓
What is the difference between a regular and an irregular polygon?
A regular polygon has:
- All sides equal in length.
- All interior angles equal.
An irregular polygon has sides and/or angles of different sizes. Most polygons you encounter in real life are irregular.
Interior angle of a regular polygon:
$$\text{Interior angle} = \frac{(n - 2) \times 180°}{n}$$
Examples:
| Regular polygon | Formula | Each interior angle |
|---|---|---|
| Equilateral triangle (3) | (3−2)×180°÷3 | 60° |
| Square (4) | (4−2)×180°÷4 | 90° |
| Regular pentagon (5) | (5−2)×180°÷5 | 108° |
| Regular hexagon (6) | (6−2)×180°÷6 | 120° |
| Regular octagon (8) | (8−2)×180°÷8 | 135° |
What is the exterior angle of a regular polygon?
At each vertex of a polygon, the exterior angle is the supplement of the interior angle (they add to 180°):
$$\text{Exterior angle} = 180° - \text{interior angle}$$
For a regular polygon, all exterior angles are equal, and there is a particularly clean result:
$$\text{Each exterior angle of a regular polygon} = \frac{360°}{n}$$
Example: Each exterior angle of a regular hexagon = 360° ÷ 6 = 60°.
This works because walking all the way round a polygon requires you to turn through a total of 360° — one full turn.
What is the difference between convex and concave polygons?
A convex polygon has all interior angles less than 180°. Every regular polygon is convex. When you stand inside a convex polygon, every side is "in front" of you.
A concave polygon (sometimes called re-entrant) has at least one interior angle greater than 180° — it has an "indentation" or "cave" in its boundary. The familiar five-pointed star (pentagram) is concave.
Key test: A polygon is convex if any line segment connecting two interior points lies entirely inside the shape.
What are the properties of regular polygons in everyday life?
- Equilateral triangle: Used in trusses, bridges, and warning signs — rigid under load.
- Square: Tiles, rooms, books — four right angles make it easy to pack together.
- Regular hexagon: Honeycomb, floor tiles, nuts and bolts — hexagons tile a plane without gaps and are structurally strong.
- Regular octagon: Stop signs in many countries.
Frequently asked questions
How do you find a missing interior angle of an irregular polygon?
Find the interior angle sum using (n − 2) × 180°, then subtract all the known angles. For example, in an irregular pentagon with angles 110°, 95°, 130°, and 85°, the fifth angle is 540° − (110° + 95° + 130° + 85°) = 540° − 420° = 120°.
Is a circle a polygon?
No. A polygon must have a finite number of straight sides. A circle has no straight sides — its boundary is a curve. You can approximate a circle with a polygon that has many sides (a polygon with 360 sides comes close), but a circle itself is not a polygon.
What is the exterior angle sum of any polygon?
The exterior angles of any convex polygon — regular or irregular — always sum to exactly 360°. This is because travelling around the perimeter of any convex polygon and turning at each vertex completes exactly one full rotation.
Why is the formula (n − 2) × 180° and not n × 180°?
If you stand at one vertex and draw diagonals to all non-adjacent vertices, you divide the polygon into (n − 2) triangles — not n. Two of the vertices at your starting point are connected by sides, not diagonals, so they don't generate new triangles. Each triangle contributes 180° to the angle sum, giving (n − 2) × 180° in total.
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