# Sum of Angles in a Polygon - Formula and Examples

TL;DR

The sum of the interior angles in a polygon equals (n−2)×180°, where n is the number of sides. This article shows why that formula works, works through pentagons, hexagons, and irregular shapes, and separates the interior-angle sum from the exterior-angle sum, which is always 360°.

## What Is The Sum of Angles In A Polygon?

The sum of the interior angles of a polygon with n sides is (n−2)×180°. That single formula covers every polygon - a triangle, a hexagon, a 100-sided shape - and it holds whether the polygon is **regular** (all sides and angles equal) or **irregular** (sides and angles of different sizes). An _interior angle_ is the angle formed inside the polygon at each vertex, between two sides that meet there.

The number never changes for a given number of sides. Bend a quadrilateral into any four-sided shape you like, and its four interior angles still add to 360°. That fixedness is what makes the formula so useful.

## Where Does The (n−2)·180° Formula Come From?

The formula drops straight out of one move: cut the polygon into triangles.

Pick any single vertex of the polygon. Draw diagonals from that one vertex to every other non-adjacent vertex. This slices the polygon into a fan of triangles that don't overlap.

- **A quadrilateral** (n=4) splits into **2 triangles**. Sum = 2×180° = 360°.
- **A pentagon** (n=5) splits into **3 triangles**. Sum = 3×180° = 540°.
- **A hexagon** (n=6) splits into **4 triangles**. Sum = 4×180° = 720°.

Count the triangles and a pattern appears: a polygon with n sides always splits into exactly n−2 triangles from a single vertex. Since each triangle's three angles sum to 180° (the [triangle sum theorem](/content/math/geometry/triangle-sum-theorem/index.html)), the whole polygon's interior angles sum to:

Sum of interior angles=(n−2)×180°

### What about one interior angle of a regular polygon?

For a **regular polygon**, every interior angle is equal, so divide the total by the number of sides:

Each interior angle=(n−2)×180°/n

A regular hexagon: 720°/6 = 120° per angle. This division step only works when the polygon is regular — for irregular polygons the total is fixed, but the individual angles can differ.

## What Is The Sum Of the Exterior Angles?

The **sum of the exterior angles of any polygon is always 360°** - no matter how many sides it has. An _exterior angle_ is the angle between one side and the extension of the next side.

Walk all the way around the boundary of the polygon and you turn through a full circle exactly once, so the turns add up to 360°. A triangle, a pentagon, a 20-sided shape - every one of them totals 360° in exterior angles. The interior sum grows with more sides; the exterior sum does not.

## Examples of the Sum of Angles in a Polygon

### Example 1

**Find the sum of the interior angles of a hexagon.**

A hexagon has n=6 sides.

Sum=(6−2)×180°=4×180°=720°.

Final answer: **720°**.

### Example 2

**A student is told a regular octagon's interior angles sum to 8 × 180° = 1440°. Is that right?**

The tempting move is to multiply the number of sides straight by 180°, because "each side has an angle."

Wrong path: 8×180°=1440°.

Check it against a shape you know. A quadrilateral has 4 sides. By the same wrong logic, 4×180°=720° - but a quadrilateral (a square, say) has four right angles summing to 4×90°=360°.

The fix is the