SECTION 04

The Sum of the Interior Angles

We already know the three angles of a triangle add up to 180°. Draw every diagonal from a single vertex of a convex polygon and it splits into triangles — counting those triangles gives the whole rule.

Splitting a polygon into triangles

Polygon with 4 sides divided into 2 triangles
4 sides → 2 triangles
Polygon with 5 sides divided into 3 triangles
5 sides → 3 triangles
Polygon with 6 sides divided into 4 triangles
6 sides → 4 triangles
Polygon with 7 sides divided into 5 triangles
7 sides → 5 triangles
Sum of interior angles = (n − 2) × 180°

Number of triangles = number of sides − 2

Reference table

Sum of the interior angles for polygons with 3 to 10 sides
PolygonSides (n)TrianglesCalculationSum
Triangle311 × 180°180°
Quadrilateral422 × 180°360°
Pentagon533 × 180°540°
Hexagon644 × 180°720°
Heptagon755 × 180°900°
Octagon866 × 180°1,080°
Nonagon977 × 180°1,260°
Decagon1088 × 180°1,440°

Worked examples

1. Hexagon ABCDEF with angles X, 90°, X, 110°, 165°, 115°. Find X.

∵ ABCDEF is a hexagon ⟹ sum = 720°

∴ X + 90 + X + 110 + 165 + 115 = 720

∴ 2X + 480 = 720 ⟹ 2X = 240

∴ X = 120

2. Pentagon ABCDE with ED ∥ BF, m(∠A) = 130°, m(∠B) = 90°. Find m(∠E).

∵ ED ∥ BF ⟹ co-interior angles sum to 180°

∴ 180° + 150° + 80° + m(∠E) = 540°

∴ 410° + m(∠E) = 540°

∴ m(∠E) = 130°