How to find the exterior angles of a polygon

If you walk around a polygon, every corner changes the direction you’re facing. Those turns give us a way to find an exterior angle, even when the polygon has several different angles.

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An exterior angle and its interior angle add to 180°

At a vertex of a convex polygon, extend one side past the vertex. The angle between that extension and the next side is an exterior angle. It forms a straight angle with the interior angle, so their measures add to 180 degrees.

The two marked angles add to 180° because they form a straight angle.

In the equilateral triangle above, all three interior angles are equal. Each measures ${60}^{°}$, so $m\angle 2={180}^{°}-{60}^{°}={120}^{°}$.

The exterior angle sum is 360°

Take one exterior angle at each vertex while walking all the way around a convex polygon in the same direction. By the end you’ve made one full turn, so the angles sum to ${360}^{°}$. For a simple concave polygon, walk counterclockwise and count left turns as positive and right turns as negative. Those signed turns also sum to ${360}^{°}$.

In a regular polygon with $n$ sides, all the exterior angles are equal, so each is $\frac{{360}^{°}}{n}$.

Use the angle sum to solve for x and y

A quadrilateral has exterior angles with numerical measures $3x+4$, $2x$, $2x-5$, and $x+15$ in degrees. Their sum gives

$\begin{array}{l}(3x+4)+2x+(2x-5)\\ +(x+15)=360\\ 8x+14=360\\ x=43.25\end{array}$

The interior angle next to the exterior angle measuring $2x$ degrees is $5y$ degrees. Since the two angles are supplementary, $5y+2\left(43.25\right)=180$. This gives $5y=93.5$, so $y=18.7$.

Add two exterior angles at a shared side

A regular heptagon and a regular pentagon share one side. Extending the shared side shows two exterior angles that together form the gap labeled z.

The dashed extension splits the gap into two adjacent exterior angles.

Use seven sides for the heptagon and five for the pentagon. Add the exact angle measures before rounding.

$z=\frac{360}{7}+\frac{360}{5}\approx 123.43$

The gap measures about ${123.43}^{°}$.

For the relationship with the angles inside a polygon, see how to find the interior angles of a polygon.

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