Regular Polygons

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.