Exterior Angles of Polygons — Definition, Formula & Examples

Q: Why do the exterior angles of any convex polygon always sum to 360°?

Imagine walking along the perimeter of the polygon. At each vertex you turn by the exterior angle. After completing the full loop, you face the same direction you started — meaning you have turned through one complete rotation, which is exactly 360°.

An exterior angle of a polygon is the angle formed between one side of the polygon and the extension of an adjacent side. For any convex polygon, the exterior angles always add up to 360°.

At each vertex of a convex polygon, the exterior angle is the supplement of the interior angle at that vertex. If the interior angle measures $\alpha$ , the corresponding exterior angle measures $180° - \alpha$ . The sum of all exterior angles (one at each vertex) of any convex polygon equals $360°$ .

Key Formula

\text{Each exterior angle of a regular polygon} = \frac{360°}{n}

Where:

$n$ = The number of sides (or vertices) of the polygon

How It Works

To find an exterior angle, extend one side of the polygon past a vertex — the angle between that extension and the next side is the exterior angle. The powerful fact here is that no matter how many sides a convex polygon has, the exterior angles always sum to exactly

360°

. For a regular polygon (all sides and angles equal), you simply divide

360°

by the number of sides to get each exterior angle. This relationship also lets you work backwards: if you know one exterior angle of a regular polygon, divide

360°

by that angle to find the number of sides.

Worked Example

Problem: Find the measure of each exterior angle of a regular octagon (8 sides).

Step 1: Write the formula for one exterior angle of a regular polygon.

\text{Exterior angle} = \frac{360°}{n}

Step 2: Substitute n = 8 for an octagon.

\text{Exterior angle} = \frac{360°}{8}

Step 3: Divide to find the answer.

\text{Exterior angle} = 45°

Answer: Each exterior angle of a regular octagon measures 45°.

Another Example

Problem: Each exterior angle of a regular polygon measures 30°. How many sides does the polygon have?

Step 1: Start with the exterior angle formula and solve for n.

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

Step 2: Substitute the given exterior angle of 30°.

n = \frac{360°}{30°} = 12

Answer: The polygon has 12 sides — it is a dodecagon.

Visualization

Why It Matters

Exterior angles appear throughout middle-school and high-school geometry courses whenever you study polygon properties or angle relationships. Engineers and architects use exterior angle calculations when designing tiled floors, gear teeth, and structures with polygonal cross-sections. Knowing that exterior angles sum to 360° also gives you a quick way to verify interior angle calculations without memorizing a separate formula.

Common Mistakes

Mistake: Confusing the exterior angle sum with the interior angle sum.

Correction: The exterior angles of any convex polygon always sum to 360°. The interior angle sum depends on the number of sides and equals (n − 2) × 180°. These are different formulas — don't swap them.

Mistake: Using the formula 360°/n on an irregular polygon to find individual exterior angles.

Correction: The formula 360°/n only gives each exterior angle when the polygon is regular (all angles equal). For irregular polygons, the individual exterior angles differ; only their sum is guaranteed to be 360°.

Related Terms

Convex — Exterior angle sum rule applies to convex polygons
Concave — Concave polygons have reflex interior angles
Diagonal of a Polygon — Another key polygon measurement
Decagon — 10-sided polygon with 36° exterior angles
Dodecagon — 12-sided polygon with 30° exterior angles
Heptagon — 7-sided polygon used in angle problems
Apothem — Related measurement in regular polygons