Hexagon Area Calculator – Regular Hexagon from Any Dimension
A regular hexagon is a six-sided polygon where every side is equal in length and every interior angle measures exactly 120°. It appears throughout nature and engineering — from honeycomb cells and basalt columns to hex-grid board games and hexagonal bolt heads. This calculator finds the enclosed area of a regular hexagon from whichever dimension you have: side length, inradius (apothem), circumradius, long diagonal, or short diagonal. All other geometric properties are derived simultaneously.
Area Formula
The standard formula for the area of a regular hexagon with side length a is:
A = (3√3 / 2) × a² ≈ 2.598076 × a²
This result comes from dividing the hexagon into six congruent equilateral triangles, each with area (√3 / 4) × a², and summing them. The factor 3√3 / 2 is a constant approximately equal to 2.598076.
Using Other Input Dimensions
If you do not know the side length directly, the calculator accepts four alternative measurements and converts them to side length first:
- Inradius (apothem) r — the perpendicular distance from the center to the midpoint of any side. Conversion:
a = 2r / √3. - Circumradius R — the distance from the center to any vertex. For a regular hexagon,
R = aexactly (see below), so no conversion is needed. - Long diagonal d — the straight line between two opposite vertices.
d = 2a, soa = d / 2. - Short diagonal dₛ — the distance between two vertices with one vertex between them.
dₛ = √3 × a, soa = dₛ / √3.
Why R = a for a Regular Hexagon
The circumradius of a regular hexagon equals its side length — a property unique to the hexagon among regular polygons. This happens because the six equilateral triangles filling the hexagon all share the center as a vertex, and each triangle has two sides equal to the circumradius and one side equal to the hexagon's side. Since the triangles are equilateral, all three sides are equal, making R = a.
Derived Properties
From side length a, all other measurements follow directly:
- Perimeter:
P = 6a - Inradius (apothem):
r = (√3 / 2) × a ≈ 0.866a - Circumradius:
R = a - Long diagonal:
d = 2a - Short diagonal:
dₛ = √3 × a ≈ 1.732a - Interior angle: 120° (each of six angles)
- Sum of interior angles: 720°
Unit Flexibility
Select any length unit for your input (mm, cm, m, km, inches, feet, or yards). You can independently choose a different unit for the area output — for example, input a side in centimeters but display the area in square meters. The calculator normalizes all values to meters internally before computing, then converts the result to your chosen area unit.
Real-World Applications
Regular hexagons appear in many practical fields. Beekeepers and biologists use hexagonal area formulas to model honeycomb cell sizes and estimate comb capacity. Architects and structural engineers design hexagonal floor tiles, paving stones, and column cross-sections. Game developers working with hex-grid maps need precise cell areas to calculate coverage and movement costs. Manufacturers rely on hex geometry for bolt and nut specifications, where the "across flats" measurement equals the long diagonal.
Accuracy
All calculations use IEEE 754 double-precision floating-point arithmetic, which provides approximately 15–16 significant digits. You can display results with 0 to 10 decimal places. The dominant source of error in practical use is the precision of your physical measurement, not the calculator's arithmetic.