MISSION · Geometry
Angles Inside a Polygon
Why every triangle you cut a shape into is worth 180°
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The idea in one sentence
Split any polygon into triangles from one corner, count them, and multiply by \(180°\) — that’s the whole interior angle rule.
\[\text{sum of interior angles} = (n - 2) \times 180°\]Where the formula comes from
Pick a corner of a pentagon and draw lines from it to every corner it isn’t already joined to. You get 3 triangles. A hexagon gives 4. An \(n\)-sided shape gives \(n - 2\), because the two corners next door to your chosen one can’t be connected to.
Every one of those triangles contributes \(180°\), and between them they account for every bit of angle inside the shape. So a pentagon’s angles total \(3 \times 180 = 540°\).
| Shape | Sides | Interior sum | Each angle if regular |
|---|---|---|---|
| Triangle | 3 | \(180°\) | \(60°\) |
| Quadrilateral | 4 | \(360°\) | \(90°\) |
| Pentagon | 5 | \(540°\) | \(108°\) |
| Hexagon | 6 | \(720°\) | \(120°\) |
| Octagon | 8 | \(1080°\) | \(135°\) |
Regular means all sides and all angles equal, and only then can you divide the sum by \(n\) to get one angle. An irregular pentagon still totals \(540°\), just shared out unevenly.
Exterior angles are the surprise
An exterior angle is the turn you make at each corner walking around the outside. Their total is \(360°\) — for every polygon, no matter how many sides.
The proof is a walk. Set off along one edge, turn at each corner, and keep going until you’re back where you started facing your original direction. You’ve made one complete revolution, so the turns total \(360°\). Nothing in that argument mentioned the number of sides.
So for a regular polygon each exterior angle is \(\frac{360°}{n}\), which is often the faster route to the interior angle — the two sit on a straight line, so they add to \(180°\).
Why only three shapes tile a floor
A regular polygon tessellates when copies fit around a point with no gaps, which needs its angle to divide exactly into \(360°\).
Triangles (\(60°\)) fit six round a point. Squares (\(90°\)) fit four. Hexagons (\(120°\)) fit three. Pentagons are \(108°\), and three leave a gap while four overlap — so regular pentagons cannot tile a floor, and no regular polygon above six sides can either. That’s why bathroom tiles and honeycombs look the way they do.
Common mix-ups
- Using \(n \times 180°\) instead of \((n-2)\). Count the triangles, not the sides.
- Dividing by \(n\) on an irregular polygon. Only regular ones share equally.
- Thinking exterior angles grow with the number of sides. They always total \(360°\).
- Mixing interior and exterior at one corner. They’re supplementary — together they make a straight line.
Check yourself
Find the interior angle sum of a decagon (10 sides), and one interior angle if it’s regular.
\((10 - 2) \times 180 = 1440°\). Regular, each angle is \(144°\) — or via the exterior angle, \(360 \div 10 = 36\), and \(180 - 36 = 144\).
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