Interior and Exterior Angle Sums and Diagonals of Polygons

The interior angles of a convex polygon with n sides add up to (n − 2) · 180°, the exterior angles (one at each vertex) always add up to 360°, and the polygon has n(n − 3)/2 diagonals. In brief: for a hexagon, the interior angle sum is 4 · 180° = 720° and there are 6 · 3/2 = 9 diagonals. In a regular hexagon each interior angle is 720°/6 = 120° and each exterior angle is 60°.

These three formulas let you find missing angles, identify polygons from their angles and count diagonals without drawing. They all follow from one idea: cutting the polygon into triangles. This guide derives the formulas, shows how to use them in problems, discusses concave polygons and gives a table of the values for n from 3 to 15.

What is the sum of the interior angles?

Choose one vertex of a convex n-gon and draw all diagonals from it. They divide the polygon into n − 2 triangles. Each triangle has an angle sum of 180°, and together their angles make up the interior angles of the polygon. So the sum is S = (n − 2) · 180°. The same formula holds for concave simple polygons, with the interior angles measured inside the polygon (a reflex angle is larger than 180°).

For a regular n-gon each interior angle is S/n = (n − 2) · 180°/n. It increases toward 180° as n grows: 60° for n = 3, 90° for n = 4, 108° for n = 5.

What is the sum of the exterior angles?

At each vertex, the exterior angle is 180° minus the interior angle. The sum over all vertices is n · 180° − (n − 2) · 180° = 360°. This is independent of n. Intuitively, if you walk around the polygon and turn at each vertex, you make one full turn of 360° by the time you return to the start. For a regular polygon each exterior angle is 360°/n.

This gives a quick way to find n from an angle: if an exterior angle of a regular polygon is 24°, then n = 360/24 = 15. If an interior angle is 156°, the exterior angle is 24° and again n = 15.

How many diagonals does a polygon have?

From each vertex, one can draw diagonals to all other vertices except itself and its two neighbors, which gives n − 3 diagonals. There are n vertices, but each diagonal is counted twice (once from each end), so the total is D = n(n − 3)/2. For n = 5, D = 5, the pentagram. For n = 8, D = 20. The formula also equals the number of pairs of vertices, n(n − 1)/2, minus the n sides, as: n(n − 1)/2 − n = n(n − 3)/2.

The number of regions into which the diagonals of a convex polygon divide its interior is more complex and depends on whether three diagonals are concurrent; for a typical polygon, the count has a closed formula that involves binomial coefficients.

How do you solve typical problems?

Example 1: missing angle

Five angles of a hexagon are 100°, 120°, 130°, 140° and 110°. Their sum is 600°. The hexagon has an interior angle sum of 720°, so the sixth angle is 120°.

Example 2: finding n from the angle sum

The interior angles of a polygon add up to 1,620°. Then (n − 2) · 180 = 1620, so n − 2 = 9 and n = 11. The polygon is an 11-gon, with 11 · 8/2 = 44 diagonals.

Example 3: finding n from the number of diagonals

A polygon has 27 diagonals. Then n(n − 3)/2 = 27, so n² − 3n − 54 = 0, which factors as (n − 9)(n + 6) = 0, so n = 9. The negative root is rejected.

Example 4: algebraic angles

The angles of a convex pentagon are x, x + 10°, x + 20°, x + 30° and x + 40°. The sum is 540°, so 5x + 100° = 540° and x = 88°. The angles are 88°, 98°, 108°, 118° and 128°. Each is between 0° and 180°, as required for a convex polygon.

Example 5: a regular polygon

Each exterior angle of a regular polygon is 18°. Then n = 360/18 = 20, and each interior angle is 162°. The interior angle sum is 18 · 180° = 3,240°. Check: 20 · 162 = 3,240 ✓.

Example 6: which regular polygon has an interior angle that is three times the exterior angle?

Let the exterior angle be e. Then the interior angle is 180° − e = 3e, so e = 45°, and n = 360/45 = 8. The polygon is a regular octagon, with an interior angle of 135°.

Table for n from 3 to 15

Where do these formulas come from?

The angle sum of a triangle is in Euclid’s Elements, and the extension to polygons follows by triangulation. The statement that the exterior angles of a convex polygon add up to 360° is an old result of Greek geometry as well. The idea of the total turning of a closed curve was generalised in the nineteenth century to smooth curves, and later became a topic in differential geometry, the theorem of turning tangents.

Where are they used today?

Architects and designers use the angle formulas for tiling floors with polygons and for cutting mitered corners: for a regular hexagonal frame, each joint is cut at 60°. Turtle graphics in programming draws a regular polygon by moving forward and turning by the exterior angle 360°/n at each corner. In computer graphics, polygons are divided into triangles by triangulation, which is the same idea as the proof of the angle sum.

Which mistakes are common?

Using n instead of n − 2

The number of triangles is n − 2, not n. A quadrilateral has 2 triangles, so 2 · 180° = 360°.

Mixing up the angle types

The sum of interior angles depends on n, but the sum of exterior angles is always 360° (one at each vertex).

Counting sides as diagonals

Diagonals join non-adjacent vertices only, so subtract the n sides, and each diagonal is counted twice if you count from each vertex.

Applying the regular polygon formula to irregular polygons

The interior angle (n − 2) · 180°/n belongs to regular polygons. The sum, however, applies to all simple polygons.