Regular Polygons

Geometry · Axiom Academy

Polygons with equal sides and equal angles — and the radius, apothem, and area formulas that symmetry unlocks. 1. What Makes a Polygon "Regular"? A polygon can have equal sides without equal angles (a rhombus), or equal angles without equal sides (a rectangle). A regular polygon insists on both at once — that double symmetry is exactly what lets us compute its parts. Drop a point at the center and connect it to the vertices. The polygon splits into n identical triangles — and each one names a part of the figure. Distance from the center to any vertex. Distance from the center to the midpoint of a side — perpendicular to that side. The length of each side — all equal in a regular polygon. The angle at the center for each triangle. The n central angles fill a full turn, so each one is 360° / n. As the side count n climbs, each interior angle grows toward (but never reaches) 180°. Each figure below shows its interior angle, which equals (n − 2)·180° / n. Every regular polygon is just n identical triangles of height a (the apothem) and base s (a side). Sum their areas and the formula falls out: total area is half the apothem times the perimeter. Let's put the formula to work on a regular hexagon. A regular hexagon has a side length of 8 cm and an apothem of 6.93 cm. Find its area. 6. Regular Polygons in Real Life The same shapes recur everywhere — chosen because their symmetry tiles cleanly, distributes force evenly, or just locks together.

This is the written version of the interactive lesson above. See the full Geometry course.