Understanding Regular Polygons — Symmetry in Every Side
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.
Properties of Common Regular Polygons
| 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° |
💡 Pro Tip — The Exterior Angle Trick: The exterior angles of any convex polygon always sum to exactly 360°, regardless of the number of sides. This means each exterior angle of a regular polygon is simply 360°/n. Walking around any convex polygon and turning at each corner, you complete exactly one full rotation. This is a powerful fact: if you know an exterior angle, you immediately know n = 360° / exterior angle.
The Apothem — A Key to Polygon Area
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)).
Interior Angles of Polygons — Why (n − 2) × 180°?
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.
Frequently Asked Questions
What is the difference between a regular and irregular polygon?
A regular polygon has all sides equal in length AND all interior angles equal in measure. An irregular polygon may have different side lengths or different angle measures (or both). This calculator applies only to regular polygons. Irregular polygon area requires different techniques such as the shoelace formula (for vertices given as coordinates) or decomposition into triangles.
Why do exterior angles always sum to 360°?
Imagine walking around the perimeter of any convex polygon. At each vertex, you turn by the exterior angle. When you return to your starting point facing the same direction, you have made exactly one full rotation — 360°. This works for any convex polygon regardless of n, making exterior angles simpler to work with than interior angles in many cases. Each exterior angle of a regular n-gon = 360°/n, and each interior angle = 180° − 360°/n = (n−2)×180°/n.
What is the apothem and how is it different from the circumradius?
The apothem (a) is the perpendicular distance from the center to the midpoint of a side — the inradius. The circumradius (R) is the distance from the center to each vertex. For a regular polygon, a < R always (except at the limit of a circle where they're equal). The relationship is: a = R × cos(π/n). For a square with side 4: a = 2 (half the side), R = 2√2 ≈ 2.83. For a hexagon with side 6: a = 6×√3/2 ≈ 5.196, R = 6.
What polygon has interior angles of 90°?
The square (n=4) is the only regular polygon with interior angles of exactly 90°. Solving (n−2)×180°/n = 90° gives n = 4. Interior angles can only be whole-degree values for specific polygons: 60° → triangle, 90° → square, 108° → pentagon, 120° → hexagon, 135° → octagon, 144° → decagon, 150° → dodecagon, 156° → 15-gon, 160° → 18-gon, 162° → 20-gon, 170° → 36-gon, 175° → 72-gon, 178° → 180-gon.
How many diagonals does a polygon have?
A polygon with n sides has n(n−3)/2 diagonals. A triangle (n=3): 0 diagonals. A square (n=4): 2 diagonals. A pentagon (n=5): 5 diagonals. A hexagon (n=6): 9 diagonals. The formula comes from choosing 2 vertices from n to connect (C(n,2) = n(n−1)/2), then subtracting the n sides that are edges, not diagonals: n(n−1)/2 − n = n(n−3)/2.
Can a polygon have an interior angle of 180° or more?
A regular polygon cannot — regular polygons are always convex, with interior angles strictly between 0° and 180°. As n approaches infinity, each interior angle approaches (but never reaches) 180°. However, irregular polygons CAN be concave, meaning some interior angles exceed 180° (called reflex angles). These are called non-convex or concave polygons. The angle-sum formula S = (n−2)×180° still holds, but individual angles may be greater than 180°.