Interior & Exterior Angles of Polygons
“A polygon's angle sum is all about 'how many triangles it splits into'.”
The formula
(n − 2) × 180°How to read it: add up all the interior angles of an n-gon and you get (n−2) times 180°
- n
- — the number of sides (or vertices) of the polygon
- n − 2
- — the number of triangles the polygon splits into
- 180°
- — the sum of interior angles of one triangle
The hook
However complex a polygon gets, the sum of its interior angles is fixed by just one number — the count of sides.
In plain words
Add up the interior angles of an n-gon and you get (n−2) × 180°. And the exterior angles of a convex polygon always add up to 360°.
The intuition
Draw diagonals from one vertex to slice the polygon into triangles. An n-gon splits into (n−2) triangles. Each triangle's interior angles sum to 180°, so the whole interior sum is 180° added (n−2) times.
How it's built
① Interior sum = number of triangles (n−2) × 180°. ② Each exterior angle is the turn you make at a vertex, so summed together they complete exactly one full turn = 360°. That 360° holds no matter how many sides.
Example
A pentagon (n=5) has an interior-angle sum of (5−2) × 180° = 3 × 180° = 540°. If it's regular, each angle is 540° ÷ 5 = 108°.
Common mistake
People assume the exterior sum varies with the number of sides, but for any convex polygon it's always 360°. Also, it's easy to mistake (n−2) for (n−1) or n.
Where it's used
Tiling (tessellation), designing angles in soccer balls and architecture, computing a regular polygon's single angle — any design where angles must fit.
Where it came from
It builds on Euclid's basic fact that a triangle's angles sum to 180°, plus the idea of slicing a polygon into triangles.
Quick check
What is the sum of the interior angles of a hexagon (n=6)?
- 540°
- 720°✓
- 900°
- 1080°
Practice
What is the sum of the interior angles of an octagon (n=8), in degrees?
Answer: 1080
- (n−2) × 180° = (8−2) × 180°
- 6 × 180° = 1080°
Takeaway: Just plug n into the formula — an octagon is worth 6 triangles.
What is the measure of one interior angle of a regular pentagon, in degrees?
Answer: 108
- Interior sum: (5−2) × 180° = 540°
- A regular pentagon has five equal angles: 540° ÷ 5 = 108°
Takeaway: For a regular polygon, divide the interior sum by the number of sides.
What is one exterior angle of a regular 12-gon? First write on your notepad what the exterior angles always sum to.
Answer: 30
- The exterior angles of a convex polygon always sum to 360°
- A regular 12-gon has equal exterior angles: 360° ÷ 12 = 30°
Takeaway: Exterior angles are easier — they always sum to 360°, so just divide by the count.