Core answer: Regular n-gon area S = n·a² ÷ [4·tan(π/n)], where a is the side length. Quick values: equilateral triangle 0.433a², square a², regular pentagon 1.720a², hexagon 2.598a², octagon 4.828a². Doubling the side quadruples the area.
The general formula
Slice the n-gon from its center into n congruent triangles, each with base a (the side) and height r (the apothem):
S = n × (a × r ÷ 2) = perimeter × apothem ÷ 2
With apothem r = a ÷ [2·tan(π/n)], combining gives:
S = n·a² ÷ [4·tan(π/n)]
Coefficient quick table
| Sides n | Name | Area formula | Coefficient (×a²) |
|---|---|---|---|
| 3 | Equilateral triangle | (√3/4)a² | 0.433 |
| 4 | Square | a² | 1.000 |
| 5 | Pentagon | ¼√(25+10√5)a² | 1.720 |
| 6 | Hexagon | (3√3/2)a² | 2.598 |
| 8 | Octagon | 2(1+√2)a² | 4.828 |
| 10 | Decagon | 7.694 | |
| 12 | Dodecagon | 3(2+√3)a² | 11.196 |
Pattern: as n grows the shape approaches a circle, and the coefficient approaches the area of the circumscribed circle.
Example 1: hexagonal floor tiles
Example: how many 20 cm-side hexagonal tiles cover 10 m²?
- Area per tile = 2.598 × 20² = 2.598 × 400 = 1,039 cm² ≈ 0.104 m²
- Count = 10 ÷ 0.104 ≈ 96 tiles; add 8% breakage → order 104
Hexagons tile the plane with zero gaps (each interior angle is 120°, three fill 360°) — which is why honeycombs use them: maximum area per unit of wall material, seamless.
Example 2: an octagonal pavilion
Example: a garden octagonal pavilion with 2.5 m sides; floor area:
S = 4.828 × 2.5² = 4.828 × 6.25 = 30.2 m²
Estimating via the circumscribed circle (R = a ÷ [2·sin(π/8)] = 2.5 ÷ 0.765 ≈ 3.27 m) would give πR² ≈ 33.6 m² — an 11% overestimate, because the octagon is smaller than its circumscribed circle.
Apothem and circumradius
| Quantity | Formula | Geometric meaning |
|---|---|---|
| Apothem r | a ÷ [2·tan(π/n)] | Center to side midpoint (inradius) |
| Circumradius R | a ÷ [2·sin(π/n)] | Center to vertex |
| Interior angle | (n−2)×180° ÷ n | 120° when n=6 |
With a known apothem, area is easiest: S = perimeter × r ÷ 2.
Handling irregular polygons
- Triangulation: split into triangles/rectangles, sum the parts
- Shoelace formula (coordinates): with vertices in order (x₁,y₁)…(xₙ,yₙ), S = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)| — the standard in surveying and CAD
- Grid estimation: overlay graph paper and count squares — best for natural boundaries (lakes, forests)
Common mistakes and myths
- "Double the sides, double the area" — no; area scales with side length squared, and the side count only shifts the coefficient. At equal perimeter, more sides means more area (approaching a circle).
- "Use the circumscribed circle as the polygon's area" — always an overestimate: a hexagon is 82.7% of its circle, an octagon 90.1%.
- "The formula takes degrees" — tan(π/n) uses radians; if your calculator is in degree mode, switch it or every result is wrong.
- "Regular polygons stop at the octagon" — any n ≥ 3 has a regular polygon; with a known side length, the general formula covers them all.
Use the [Polygon Area Calculator](/c/geometry/polygon) for area, apothem, and interior angles from sides and side length, and the [Circle Area Calculator](/c/geometry/circle) for the bounding-circle comparison.