Area, perimeter, angles, and apothem for any regular polygon — with step-by-step solutions and a live shape preview.
The interior angle sum formula S = (n−2) × 180° has a clean geometric proof: any polygon with n sides can be divided into exactly (n−2) triangles by drawing diagonals from one vertex. Since each triangle contains 180°, the total is (n−2) × 180°. For a hexagon: (6−2) × 180° = 720°, so each interior angle = 120°. Three hexagons meeting at a point sum to 360° exactly — which is why hexagons tile perfectly. The apothem formula a = s / (2 × tan(π/n)) comes from the right triangle formed by the apothem, half a side, and the circumradius: tan(π/n) = (s/2) / a. Remember: the apothem (center to side midpoint) is always shorter than the circumradius (center to vertex).
A regular polygon has all sides equal in length and all interior angles equal in measure. This perfect symmetry makes calculation straightforward: every property can be derived from just two inputs — the number of sides (n) and the side length (s). From these, you can find area, perimeter, interior and exterior angles, and the apothem (the distance from center to the middle of any side).
The most famous regular polygons have specific names: equilateral triangle (3), square (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10), dodecagon (12), and icosagon (20). As n increases toward infinity, a regular polygon approaches a circle — a fact exploited in ancient approximations of π.
Regular polygons appear everywhere in nature and architecture. Honeybee combs use regular hexagons because they tile the plane perfectly while minimizing material per unit area — the "honeycomb conjecture," proven formally in 1999. Basalt columns from volcanic cooling often form hexagons for the same energy-minimization reason. Stop signs are regular octagons; soccer balls are assembled from regular pentagons and hexagons.
| Polygon | Sides | Interior Angle | Sum of Angles | Exterior Angle |
|---|---|---|---|---|
| Triangle | 3 | 60° | 180° | 120° |
| Square | 4 | 90° | 360° | 90° |
| Pentagon | 5 | 108° | 540° | 72° |
| Hexagon | 6 | 120° | 720° | 60° |
| Octagon | 8 | 135° | 1080° | 45° |
| Decagon | 10 | 144° | 1440° | 36° |
| Dodecagon | 12 | 150° | 1800° | 30° |
The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of any side. It's the "inradius" — the radius of the largest circle that fits inside the polygon. The area formula A = ½ × Perimeter × Apothem is a generalization of the triangle area formula (½ × base × height), where the entire perimeter acts as the "base" and the apothem acts as the "height" when you think of the polygon as a collection of triangles fanning out from the center.
The apothem formula a = s / (2 × tan(π/n)) comes from trigonometry. Each of the n triangles formed from the center has a base of s and two sides of length R (circumradius). The apothem is the height of each triangle, found using: tan(π/n) = (s/2) / a, which rearranges to a = (s/2) / tan(π/n) = s / (2tan(π/n)).
Graph polygon vertices, compute trig values, and verify angle calculations for any polygon.
View on Amazon →Construct regular polygons accurately using a compass and 360° protractor.
View on Amazon →Master polygon proofs, angle theorems, and area formulas with worked examples.
View on Amazon →The formula for the sum of interior angles — S = (n − 2) × 180° — comes from triangulation. Any polygon with n sides can be divided into exactly (n − 2) non-overlapping triangles by drawing diagonals from one vertex. Since each triangle contains 180°, the total interior angle sum is (n − 2) × 180°.
For a triangle (n=3): S = 1 × 180° = 180° ✓. For a quadrilateral (n=4): S = 2 × 180° = 360° ✓. For a hexagon (n=6): S = 4 × 180° = 720°, so each interior angle of a regular hexagon = 720°/6 = 120° — which is why hexagons tile perfectly (3 × 120° = 360°, filling exactly one point where three hexagons meet).
This triangulation approach also proves the formula works for irregular polygons — any polygon with n sides, regardless of shape, has the same sum of interior angles as a regular n-gon. Only the individual angle values differ.