Interior and Exterior Angles of a Polygon

Interior and Exterior Angles of a Polygon — Moosa Academy

A polygon with more sides has larger interior angles — a triangle's corners are 60°, an octagon's are 135°. Yet the exterior angles of every polygon add to the same total, no matter how many sides it has. Two short rules explain both facts.

Theorem The two rules

Sum of the interior angles:

(n - 2) \times 180^\circ

Sum of the exterior angles:

360^\circ

, for every polygon

Here $n$ is the number of sides. In a regular polygon all angles are equal, so each interior angle is the sum divided by $n$ , and each exterior angle is $360^\circ \div n$ .

Concept Why the interior rule works

Pick one corner and draw every diagonal from it. A pentagon splits into 3 triangles, a hexagon into 4, and in general an

n

-sided polygon into

n - 2

triangles.

Each triangle contributes 180°, and together they cover the whole interior exactly once. So the total is $(n - 2) \times 180^\circ$ .

Concept Why the exterior sum is always 360°

Imagine walking once around the outside of the polygon. At every corner you turn by the exterior angle at that corner. By the time you are back where you started, facing the way you first faced, you have turned through one full circle:

\text{total turning} = 360^\circ

The number of corners never enters this argument. A triangle turns 120° three times; a dodecagon turns 30° twelve times. Both journeys add up to one full circle, which is why the exterior sum never changes.

Example A regular octagon

Find each interior angle and each exterior angle when $n = 8$ .

Interior sum =

(8 - 2) \times 180^\circ = 1080^\circ

Each interior angle =

1080^\circ \div 8 = 135^\circ

Each exterior angle =

360^\circ \div 8 = 45^\circ

Check: an interior angle and its exterior angle sit on a straight line, and $135^\circ + 45^\circ = 180^\circ$ . That check works for every regular polygon.

Note The common polygons

Shape	n	Interior sum	Each interior	Each exterior
Triangle	3	180°	60°	120°
Square	4	360°	90°	90°
Pentagon	5	540°	108°	72°
Hexagon	6	720°	120°	60°
Octagon	8	1080°	135°	45°

Reading down the last two columns: as $n$ grows the interior angle climbs towards 180° and the exterior angle shrinks towards 0°. The polygon is slowly turning into a circle.

Summary

The interior angles of an n-sided polygon add to (n − 2) × 180°.
This is because the polygon splits into n − 2 triangles from a single corner.
The exterior angles always add to 360°, whatever n is.
That is one full turn made by walking once around the shape.
In a regular polygon divide each sum by n to get a single angle.
An interior angle and its exterior angle always add to 180°.