Interior Angles of Polygons

The sum of the interior angles of any polygon depends only on the number of sides. For an n-sided polygon the sum is (n − 2) × 180°. Regular polygons have all angles equal, so each interior angle = sum ÷ n.

What is a Polygon?

A polygon is a closed flat shape with straight sides.

Sides	Name
3	Triangle
4	Quadrilateral
5	Pentagon
6	Hexagon
8	Octagon
10	Decagon

Interior Angle Sum Formula

Interior Angle Sum of a Polygon

\text{Sum of interior angles} = (n - 2) \times 180°

n = number of sides

Where does this come from? Any polygon can be split into triangles by drawing diagonals from one vertex. An $n$ -sided polygon splits into $(n-2)$ triangles, each with 180°.

Shape	$n$	$(n-2) \times 180°$
Triangle	3	180°
Quadrilateral	4	360°
Pentagon	5	540°
Hexagon	6	720°

Each Angle in a Regular Polygon

A regular polygon has all sides equal and all angles equal.

$\text{Each interior angle} = \frac{(n-2) \times 180°}{n}$

Each angle of a regular hexagon

$\text{Each angle} = \frac{(6-2) \times 180°}{6} = \frac{720°}{6} = 120°$

Exterior Angles

An exterior angle is formed by extending one side. For any polygon:

$\text{Sum of exterior angles} = 360° \quad \text{(always)}$

For a regular polygon, each exterior angle = $360° \div n$ .

Tip

Interior angle + exterior angle = 180° (they are on a straight line).

Real world: hexagonal floor tiles

Regular hexagonal tiles have interior angles of $\frac{(6-2) \times 180°}{6} = 120°$ .

Three tiles meeting at a point give $3 \times 120° = 360°$ — exactly a full turn. That is why hexagons tessellate perfectly with no gaps.

Real world: stop sign

A stop sign is a regular octagon. Each interior angle is $\frac{(8-2) \times 180°}{8} = 135°$ .

Its distinctive shape is recognisable from any direction — the polygon angle formula explains its geometry.

Real world: football (soccer ball)

A traditional football combines regular pentagons (interior angle 108°) and hexagons (120°). These specific angles allow the flat panels to curve into a near-spherical shape when stitched together.

Practice Problems

Find the sum of interior angles of an octagon (8 sides).

Show answer

(8 − 2) × 180° = 6 × 180° = 1080°

Each interior angle of a regular polygon is 135°. How many sides does it have?

Show answer

Exterior angle = 180° − 135° = 45°. Number of sides = 360° ÷ 45° = 8 sides (regular octagon).

Three angles of a quadrilateral are 85°, 110°, and 95°. Find the fourth angle.

Show answer

Sum = 360°. Fourth angle = 360° − 85° − 110° − 95° = 70°.