An exterior angle is the turn from one side to the extension of the next. Walking once around a convex polygon makes one complete turn: one exterior angle at each vertex, taken consistently in the same direction, totals 360. At each vertex, interior angle + exterior angle =180. For a regular n-gon, each exterior angle is 360 n, and n= 360 exterior angle. **Regular polygon** | **Interior angle** | **Exterior angle** Pentagon | 108 | 72 Hexagon | 120 | 60 Octagon | 135 | 45 Decagon | 144 | 36 Example A regular polygon has interior angle 156. Its exterior angle is 24, so it has 360 24=15 sides. The side count must be an integer at least 3.

An exterior angle is the turn from one side to the extension of the next. Walking once around a convex polygon makes one complete turn: one exterior angle at each vertex, taken consistently in the same direction, totals \(360^\circ\).

At each vertex, interior angle + exterior angle \(=180^\circ\). For a regular \(n\)-gon, each exterior angle is \(\dfrac{360^\circ}{n}\), and \(n=\dfrac{360^\circ}{\text{exterior angle}}\).

A regular polygon has interior angle \(156^\circ\). Its exterior angle is \(24^\circ\), so it has \(360\div24=15\) sides. The side count must be an integer at least 3.

Regular polygon	Interior angle	Exterior angle
Pentagon	\(108^\circ\)	\(72^\circ\)
Hexagon	\(120^\circ\)	\(60^\circ\)
Octagon	\(135^\circ\)	\(45^\circ\)
Decagon	\(144^\circ\)	\(36^\circ\)

Exterior angles and regular polygons

Exterior angles and regular polygons