Math Core

Triangles have an angle sum of $180^\circ$ , no matter their shape. Polygons with more sides have a fixed angle sum too, and it depends only on how many sides there are. In this lesson you'll find that sum, use it to solve for missing angles, and discover a surprising fact about exterior angles.

Polygon vocabulary

A polygon is a closed plane figure made of three or more segments, called sides. Each side meets exactly two other sides, one at each endpoint, and the sides meet only at their endpoints. Each endpoint is a vertex. A segment that joins two vertices that are not next to each other is a diagonal.

Polygons are named by their number of sides:

Sides	Name	Sides	Name
3	triangle	8	octagon
4	quadrilateral	9	nonagon
5	pentagon	10	decagon
6	hexagon	12	dodecagon
7	heptagon	$n$	$n$ -gon

A polygon is convex if no line that contains a side passes through the inside of the polygon. Every interior angle of a convex polygon is less than $180^\circ$ . A polygon that "caves in" somewhere is concave. In this unit, "polygon" means a convex polygon unless it says otherwise.

Definition

Regular polygon

A polygon is regular if it is both equilateral (all sides congruent) and equiangular (all interior angles congruent).

A rhombus is equilateral but not always equiangular, and a rectangle is equiangular but not always equilateral. Neither one is regular unless it is a square.

The interior angle sum

Pick one vertex of a polygon and draw every diagonal from it. The diagonals split the polygon into triangles that don't overlap, and together the angles of those triangles make up exactly the interior angles of the polygon.

Two diagonals from vertex A split pentagon ABCDE into three triangles.

How many triangles do you get? From one vertex you can't draw a diagonal to that vertex itself or to its two neighbors, so an $n$ -gon has $n - 3$ diagonals from each vertex. Those diagonals make $n - 2$ triangles.

Polygon	Sides $n$	Triangles $n - 2$	Angle sum
triangle	3	1	$180^\circ$
quadrilateral	4	2	$360^\circ$
pentagon	5	3	$540^\circ$
hexagon	6	4	$720^\circ$

Each triangle contributes $180^\circ$ , which gives the general rule.

Polygon interior angles theorem

The sum of the interior angle measures of a convex $n$ -gon is

(n - 2) \cdot 180^\circ.

In a regular $n$ -gon every interior angle has the same measure, so each one is $\dfrac{(n - 2) \cdot 180^\circ}{n}$ .

Worked example: An octagon

Find the sum of the interior angles of an octagon, and the measure of each interior angle of a regular octagon.

An octagon has $n = 8$ sides:

(8 - 2) \cdot 180^\circ = 6 \cdot 180^\circ = 1080^\circ.

A regular octagon has eight congruent angles, so each one measures $1080^\circ \div 8 = 135^\circ$ . (A stop sign has $135^\circ$ corners.)

Worked example: Solving for a missing angle

The interior angles of a pentagon measure $95^\circ$ , $120^\circ$ , $110^\circ$ , $x^\circ$ and $(x + 15)^\circ$ . Find $x$ and the two unknown angles.

A pentagon's angles add to $(5 - 2) \cdot 180^\circ = 540^\circ$ :

\begin{aligned} 95 + 120 + 110 + x + (x + 15) &= 540 \\ 2x + 340 &= 540 \\ 2x &= 200 \\ x &= 100 \end{aligned}

The unknown angles are $100^\circ$ and $115^\circ$ . Check: $95 + 120 + 110 + 100 + 115 = 540$ .

You can also run the theorem backward. If the angle sum of a polygon is $1800^\circ$ , then $(n - 2) \cdot 180 = 1800$ , so $n - 2 = 10$ and $n = 12$ . The polygon is a dodecagon.

Common mistake

The formula uses $n - 2$ , not $n$ . A common error is to multiply the number of sides by $180^\circ$ . Test yourself on a shape you know: a quadrilateral has $n = 4$ , and $(4 - 2) \cdot 180^\circ = 360^\circ$ is right, while $4 \cdot 180^\circ = 720^\circ$ is not.

Exterior angles

At each vertex, extend one side past the vertex. The angle between the extension and the next side is an exterior angle. An interior angle and its exterior angle form a linear pair, so they add to $180^\circ$ .

Now add up one exterior angle at each vertex of an $n$ -gon. The $n$ linear pairs total $180n$ degrees. The interior angles account for $(n - 2) \cdot 180 = 180n - 360$ degrees of that, so the exterior angles account for the rest:

180n - (180n - 360) = 360.

The $n$ cancels, so the answer is the same for every polygon.

Polygon exterior angles theorem

If you take one exterior angle at each vertex of a convex polygon, their measures add to $360^\circ$ , whatever the number of sides.

In a regular $n$ -gon each exterior angle measures $\dfrac{360^\circ}{n}$ .

Here is a way to picture it. Walk all the way around the edge of a polygon. At each corner you turn through the exterior angle. When you get back to the start you are facing the way you began, so you have turned exactly one full circle, $360^\circ$ .

Worked example: Finding the number of sides

Each interior angle of a regular polygon measures $156^\circ$ . How many sides does it have?

Each exterior angle measures $180^\circ - 156^\circ = 24^\circ$ . The exterior angles add to $360^\circ$ , so

n = \frac{360^\circ}{24^\circ} = 15.

The polygon has $15$ sides. Check with the interior formula: $\dfrac{13 \cdot 180^\circ}{15} = \dfrac{2340^\circ}{15} = 156^\circ$ .

Tip

For regular polygons, the exterior angle is usually the fastest route. Find $360^\circ \div n$ first, then subtract from $180^\circ$ to get the interior angle. It also tells you when a regular polygon can't exist: an interior angle of $130^\circ$ would need exterior angles of $50^\circ$ , and $360 \div 50 = 7.2$ isn't a whole number of sides.