Triangles & Polygons (Years 7–11) — Maths

A polygon is any closed shape with straight sides — triangles, squares, pentagons, the honeycomb hexagons bees build, the STOP sign octagon on the road. This topic is about working out a polygon's angles: both the interior ones at each corner and the exterior ones formed by extending a side.

Why this matters: tilers, quilters and games designers all need to know which shapes fit together without gaps. Whether regular hexagons tessellate, why a football mixes pentagons and hexagons, how a satellite dish or a bolt head is shaped — these are polygon-angle questions. Learn the two formulas below and you can find any angle of any regular shape in seconds.

A pentagon splits into 3 triangles → angles sum to 3 × 180° = 540°

Interior angles (the corners)

Every triangle's interior angles total 180° — the building block.

Here is the key idea, shown in the picture: pick one corner of a polygon and draw lines to every other corner. You slice the shape into triangles, and each triangle is 180°. A shape with n sides always splits into (n − 2) triangles — a pentagon into 3, a hexagon into 4 — which gives the formula:

Sum of interior angles = (n − 2) × 180°

Hexagon (6 sides): (6 − 2) × 180 = 4 × 180 = 720°. Pentagon (5 sides): (5 − 2) × 180 = 540°.

Regular polygons and exterior angles

Interior and exterior sit on a straight line, so they add to 180°.

A regular polygon has all sides and all angles equal — a square, an equilateral triangle, a regular hexagon. Now think of walking once around the outside of the shape: at each corner you turn by the exterior angle, and by the time you are back where you started you have turned one full circle. So the exterior angles always add to 360°, however many sides there are. That gives:

Each exterior angle = 360° ÷ n · Each interior angle = 180° − exterior

The interior and its exterior sit on a straight line together, which is why they pair up to 180°.

Regular pentagon (5 sides): exterior = 360 ÷ 5 = 72°, so each interior = 180 − 72 = 108°.

The exterior angles of any polygon always total one full turn.

Useful backwards too: if you know one exterior angle, the number of sides is 360 ÷ exterior. An exterior of 40° means 360 ÷ 40 = 9 sides. And a stated interior angle gives the exterior first (180 − interior), then the sides.

Regular shape	Sides n	Exterior (360/n)	Interior
Triangle	3	120°	60°
Square	4	90°	90°
Pentagon	5	72°	108°
Hexagon	6	60°	120°
Octagon	8	45°	135°

Common mistake: using (n − 2) × 180 to find one angle. That formula gives the total of all the corners. For a single interior angle of a regular shape, either do the exterior route (180 − 360/n) or divide the total by n.

Memory hook: Interior total = (n − 2) × 180. For anything regular, go through the exterior: it is just 360 ÷ n, and interior = 180 − exterior.

When the angles are written with a letter

The angle sums still hold, so turn them into an equation.

Worked example. The exterior angles of a pentagon are 4x, 3x, 2x, 2x and x degrees. The exterior angles of any polygon add to 360°:

4x + 3x + 2x + 2x + x = 360
Collect the terms: 12x = 360
Divide by 12: x = 30

Check: 120 + 90 + 60 + 60 + 30 = 360. ✓