Regular Polygons

The most even shapes there are

Look at a stop sign, a honeycomb, the head of a bolt, a snowflake, the panels on a football. Over and over you meet shapes that are perfectly even — every side the same, every corner the same. These are the regular polygons, and once you learn their one rule and two little formulas, you can work out every angle in any of them, no protractor needed.

A stop sign is a regular octagon (8 sides). Honeycomb is built from regular hexagons (6 sides). A square is the regular 4-gon. Even the humble equilateral triangle is the regular 3-gon. They all obey the same tidy maths, which is what makes them so useful — and so beautiful.

All sides, and all angles, equal

A polygon is regular when it is as even as it can be: every side the same length and every interior angle the same size. Both conditions matter — a rectangle has all angles equal but not all sides, and a rhombus has all sides equal but not all angles, so neither is regular.

Once a polygon is regular, its angles are fixed by a single number — how many sides n it has. Walk all the way around the outside and you turn through a full 360^\circ, shared equally among the n corners, so each exterior angle is 360^\circ / n. The interior angle is whatever is left on the straight line beside it.

\text{interior} = 180^\circ - \frac{360^\circ}{n} = \frac{(n-2)\times 180^\circ}{n}

In a regular polygon with n sides:

all sides are equal and all interior angles are equal;
each exterior angle is \dfrac{360^\circ}{n};
each interior angle is 180^\circ - \dfrac{360^\circ}{n} = \dfrac{(n-2)\times 180^\circ}{n};
so the equilateral triangle has 60^\circ angles, the square 90^\circ, the regular pentagon 108^\circ and the regular hexagon 120^\circ.

Meet the family

Step through four regular polygons. Each one has its vertices spaced evenly around a circle, so all its sides match and all its angles match — and the interior angle grows as the number of sides grows.

Worked example 1 — hexagon and octagon

Regular hexagon (n = 6). The quickest route is the exterior angle first:

\text{exterior} = \frac{360^\circ}{6} = 60^\circ, \qquad \text{interior} = 180^\circ - 60^\circ = 120^\circ.

So every corner of a honeycomb cell is 120^\circ.

Regular octagon (n = 8), the stop-sign shape:

\text{exterior} = \frac{360^\circ}{8} = 45^\circ, \qquad \text{interior} = 180^\circ - 45^\circ = 135^\circ.

Worked example 2 — working backwards

Here is a puzzle the formula solves instantly. A regular polygon has each exterior angle equal to 40^\circ. How many sides does it have?

The exterior angles always add up to 360^\circ and here each one is 40^\circ, so just count how many fit:

n = \frac{360^\circ}{40^\circ} = 9.

It is a regular nonagon (9 sides). Going backwards is often easier than going forwards, because dividing into 360 is a single clean step.

Worked example 3 — check it two ways

A good habit: find an interior angle by both formulas and make sure they agree. Take a regular decagon (n = 10).

Exterior route: 180^\circ - \dfrac{360^\circ}{10} = 180^\circ - 36^\circ = 144^\circ.
Angle-sum route: \dfrac{(10-2)\times 180^\circ}{10} = \dfrac{1440^\circ}{10} = 144^\circ.

Both give 144^\circ. They must — they are the same formula wearing two different disguises, which is exactly why you can trust either one.

"Regular" is a two-part promise: equal sides AND equal angles. Miss half of it and the shape is not regular, even though it looks tidy:

A rhombus (think of a leaning diamond) has all four sides equal — but its angles are not all equal. Not regular.
A rectangle has all four angles equal (all 90^\circ) — but its sides are not all equal. Not regular.
Only when both hold — as in the square — is a four-sided shape regular.

Try to tile a floor using copies of a single regular polygon with no gaps and no overlaps. It turns out only three shapes can do it alone: the equilateral triangle, the square, and the regular hexagon. The reason is pure angle arithmetic — the interior angles have to divide evenly into 360^\circ around each meeting point, and only 60^\circ, 90^\circ and 120^\circ do (a pentagon's 108^\circ never fits).

Of those three, the hexagon wins a hidden contest: it wraps the most space inside the least wall. That is why bees build hexagonal honeycomb — it stores the most honey for the least wax. Insects were "solving" a geometry optimisation problem millions of years before humans managed to prove it was the best answer.