Sum of Angles in a Polygon - Formula and Examples

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.

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), 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