Angles, polygons, and why the rules are not arbitrary
Nearly all of it follows from one fact about triangles.
Geometry at school can feel like a list of facts to memorise. Most of it follows from a single fact plus a bit of reasoning, and it is much easier to carry that way.
The fact everything comes from
The angles in a triangle add to 180 degrees.
From this everything else about polygons follows, because any polygon can be cut into triangles from one corner. A quadrilateral splits into 2 triangles, so its angles total 360. A pentagon splits into 3, giving 540. In general an n-sided polygon splits into n-2 triangles, so its interior angles sum to (n-2) x 180.
- Triangle: 180
- Quadrilateral: 360
- Pentagon: 540
- Hexagon: 720
Nothing there needs memorising once you can picture the triangles.
Regular polygons
A regular polygon has all sides and angles equal, so each interior angle is the total divided by n:
- Equilateral triangle: 180/3 = 60
- Square: 360/4 = 90
- Regular pentagon: 540/5 = 108
- Regular hexagon: 720/6 = 120
There is a neater route to the same numbers. Walk around the outside of any polygon and you turn through one full circle, so the exterior angles always sum to 360 regardless of the number of sides. Each exterior angle of a regular n-gon is 360/n, and the interior angle is 180 minus that.
Why only three shapes tile a floor
To tile without gaps, the angles meeting at a point must add to exactly 360. So a regular polygon tiles only if its interior angle divides 360 exactly.
Only three regular polygons tile the plane on their own: triangles, squares and hexagons. Every regular tiling you have ever seen - floor tiles, honeycomb, graph paper - is one of these three, and the reason is nothing more than which numbers divide 360.
Bees build hexagons because hexagons tile with the least wall material per unit of area. That is not instinct being clever - it is the only efficient option available, and evolution found it.
Where 360 came from
The Babylonians counted in base 60, and 360 is 6 x 60. It also happens to be close to the number of days in a year, and it is divisible by 24 different numbers - which is why halves, thirds, quarters, fifths, sixths, eighths, tenths and twelfths of a circle are all whole numbers of degrees.
So it is arbitrary in origin and convenient in practice. Mathematicians mostly use radians instead, where a full turn is 2 pi, because that makes calculus work cleanly. Both are conventions; neither is more true.
Angles in reasoning puzzles. Rotation questions usually use 90, 180 and 270 because those map a square grid onto itself. If a puzzle rotates by 45 or 60, the underlying grid is probably triangular or hexagonal - worth noticing, because it tells you what symmetry to expect.
The three facts worth having instantly
- Angles in a triangle: 180. Everything else follows.
- Exterior angles of any polygon: 360, always.
- Angles at a point: 360. Angles on a straight line: 180.
With those three you can derive the rest at the moment you need it, which is more reliable than remembering a table you last saw a year ago.