a regular polygon

Some polygons are perfectly even — every side the same length, every corner the same angle — like the equilateral triangle, the square, or the honeycomb hexagon. A figure that is even in both ways at once is a regular polygon. The regular n-gons are the most symmetric polygons: spin one by the right fraction of a turn and it lands exactly on itself.

Precisely, a regular polygon is a polygon that is both equilateral (all sides equal) and equiangular (all interior angles equal). Both conditions are needed: a rhombus is equilateral but usually not equiangular, and a rectangle is equiangular but usually not equilateral; neither is regular unless it is a square. Every regular n-gon has a centre — a single point equidistant from all vertices and also from all sides — so it has both a circumscribed circle through the vertices and an inscribed circle touching the sides. The central angle, the angle at the centre subtended by one side, is 360/n degrees; each interior angle is (n-2)·180/n degrees, and each exterior angle is 360/n degrees.

Regular polygons sit at the heart of symmetry and construction. They tile the plane only for n = 3, 4, 6 (their interior angles 60, 90, 120 divide 360 evenly); and a deep result of Gauss tells exactly which regular n-gons can be drawn with compass and straightedge — those for which n is a power of 2 times distinct Fermat primes (so the 17-gon is constructible, but the regular 7-gon and 9-gon are not). One caution: 'regular' is stronger than 'equilateral'. Equilateral alone is not enough in four or more sides — the rhombus is the standard counterexample — and in school problems you must verify both the sides and the angles before claiming a polygon is regular.