Polygon Angle-Sum Theorems

By the end of this lesson, you will be able to find the sum of interior angles of any polygon using the formula $(n-2) \cdot 180°$, find individual interior angles of regular polygons, and find exterior angle sums.

The Big Idea

Every polygon can be divided into triangles by drawing diagonals from one vertex. Since each triangle contains 180°, the total interior angle sum is simply the number of triangles times 180°. This elegant argument works for any convex polygon and gives us a formula that applies from triangles to thousand-sided polygons.

Interior Angle Sum Formula

For a convex polygon with $n$ sides:

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

Why: Draw all diagonals from one vertex. An $n$-gon becomes $(n-2)$ triangles. Each triangle = 180°. Total = $(n-2) \times 180°$.

Polygon Angle Sums

Polygon	$n$	Interior angle sum
Triangle	3	$180°$
Quadrilateral	4	$360°$
Pentagon	5	$540°$
Hexagon	6	$720°$
Octagon	8	$1080°$
Decagon	10	$1440°$
$n$-gon	$n$	$(n-2) \cdot 180°$

Regular Polygons

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

Interior angle of a regular $n$-gon:

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

Exterior Angle Sum

The sum of the exterior angles of any convex polygon is always 360°, regardless of the number of sides.

One exterior angle per vertex, each formed by extending one side.

For a regular $n$-gon:

$$\text{Each exterior angle} = \frac{360°}{n}$$

And: interior angle + exterior angle = 180° (linear pair).

Example 1 — Interior Angle Sum

Find the sum of interior angles of a 9-gon (nonagon).

$(9 - 2) \times 180° = 7 \times 180° = 1260°$

Example 2 — Regular Polygon Interior Angle

Find each interior angle of a regular 12-gon.

$$\frac{(12-2) \times 180°}{12} = \frac{1800°}{12} = 150°$$

Each exterior angle: $180° - 150° = 30°$. Check: $30° \times 12 = 360°$ ✓