Hexagon Calculator

Compute every essential dimension of a regular hexagon: circumradius, inradius, area, perimeter, short & long diagonals, and interior angles. The regular hexagon is unique because its side length equals the circumradius — a property leveraged in nature and engineering. Instant results, live diagram.

Unit displayed alongside results; area shown in square units.
e.g., 1, 2.5, 0.8 — all derived properties scale accordingly.
a = 1
a = 2
a = 3
a = 4.5
a = 10
Privacy first: All computations happen locally in your browser. Zero data is transmitted.

What is a Regular Hexagon?

A regular hexagon is a six‑sided polygon with all sides equal and all interior angles equal (120° each). It is one of the most geometrically rich shapes: the side length equals the circumradius, making it uniquely constructible with a compass and straightedge. Regular hexagons tile the plane without gaps — a property famously exploited by honeybees to maximize storage efficiency. The hexagon appears across crystallography, mechanical engineering, and modern parametric design.

Essential formulas for regular hexagon (side a)

Circumradius: \( R = a \)
Inradius (apothem): \( r = \frac{\sqrt{3}}{2} a \)
Area: \( A = \frac{3\sqrt{3}}{2} a^2 \)
Short diagonal (spanning 2 sides): \( d_2 = \sqrt{3}\,a \)
Long diagonal (spanning 3 sides): \( d_3 = 2a \)

Historical & Mathematical Significance

The regular hexagon fascinated ancient Greek geometers. Euclid’s Elements Book IV demonstrates how to inscribe a regular hexagon in a circle using only the radius. Because the chord of 60° equals the radius, construction is elementary. In nature, hexagonal patterns minimize surface energy: graphene, basalt columns, and snowflakes reveal hexagonal symmetry. The Swiss mathematician Jakob Steiner proved that among all planar figures of equal area, the hexagon (as part of honeycomb conjecture) offers minimal perimeter for tiling — a theorem finally proven in 1999 by Thomas Hales.

Why Use a Dedicated Hexagon Calculator?

  • Instant precision – Avoid manual square‑root approximations; get exact floating‑point accuracy.
  • Educational depth – Visualize how area and diagonal lengths change with side length.
  • Engineering & design – Perfect for hexagonal tiles, bolts, nuts, or cellular layouts.
  • Parametric modeling – Quickly generate reference values for CAD or 3D printing.

Step-by-step derivation

  1. A regular hexagon can be partitioned into 6 equilateral triangles of side length a (since central angle = 60°).
  2. Circumradius R equals the side of those equilateral triangles → R = a.
  3. The inradius (apothem) r is the altitude of each equilateral triangle: r = a·sin(60°) = a·√3/2.
  4. Area = 6 × (area of equilateral triangle) = 6 × (√3/4 a²) = (3√3/2) a².
  5. Diagonals: d₂ connects vertices with one vertex between them → chord of 120°: d₂ = 2R sin(120°/2) = 2a sin(60°) = √3 a. The long diagonal d₃ passes through the center: d₃ = 2R = 2a.

Example Values (side a = 1, unitless)

Side a Circumradius R Inradius r Area A Short diag. d₂ Long diag. d₃
1 1.000000 0.866025 2.598076 1.732051 2.000000
2 2.000000 1.732051 10.392304 3.464102 4.000000
3 3.000000 2.598076 23.382685 5.196152 6.000000
5 5.000000 4.330127 64.951905 8.660254 10.000000
10 10.000000 8.660254 259.807621 17.320508 20.000000
Real‑world application: Hexagonal bolt design

A mechanical engineer needs to design a stainless steel hexagonal bolt head with side length 8 mm. Using this calculator, circumradius = 8 mm (directly affects wrench size), inradius ≈ 6.928 mm, area ≈ 166.277 mm². The short diagonal (≈13.856 mm) determines the minimal clearance. The tool provides immediate, reliable numbers for manufacturing specs, avoiding miscalculations that could lead to improper fastening.

Surprising fact: Hexagon and π

Archimedes used inscribed and circumscribed hexagons to bound π. Starting with a regular hexagon, he progressively doubled sides to obtain his famous approximation 3 + 10/71 < π < 3 + 1/7. The hexagon remains a cornerstone in numerical approximation of π and trigonometric tables.

Core JavaScript Implementation (excerpt) 

const n = 6;
const angle = Math.PI / n;   // 30°
let a = parseFloat(side);
let R = a;                             // circumradius
let r = a * Math.cos(angle);           // = a * √3/2
let area = (3 * Math.sqrt(3) / 2) * a * a;
let perimeter = 6 * a;
let d2 = 2 * R * Math.sin(2 * angle);   // √3 a
let d3 = 2 * R * Math.sin(3 * angle);   // 2a

Common Misconceptions

  • “All hexagon diagonals are equal.” – No, there are two distinct diagonal lengths (short and long).
  • “Inradius equals side length.” – Inradius = a·√3/2 ≈ 0.866a, not a.
  • “Area scales linearly with side.” – Area scales quadratically: doubling side quadruples area.
  • “Regular hexagon cannot tile the plane.” – Incorrect; it is one of three regular polygons that tessellate (triangle, square, hexagon).

Applications across fields

  • Architecture – Hexagonal floor plans, facades, and dome structures maximize strength and aesthetics.
  • Biology – Honeycomb, compound eyes, and carbon nanotubes exhibit hexagonal patterns.
  • Games & graphics – Hex grids for strategy games (Civ, board games) offer equidistant movement.
  • Materials science – Graphene’s carbon atoms form a hexagonal lattice, enabling revolutionary electronics.

Authoritative foundation – This tool implements rigorous Euclidean geometry and trigonometry verified against standard references (Weisstein, Eric W. “Hexagon.” MathWorld; “CRC Standard Mathematical Tables”). Calculations use IEEE 754 double precision. Content reviewed by the GetZenQuery Tech  team. Last reviewed: March 2026.

Frequently Asked Questions

Because the central angle is 60°, forming equilateral triangles between center and vertices. Each triangle has two radii and one side — all equal. Therefore R = a.

Short diagonal skips one vertex (connects vertices with one vertex between), length = √3·a. Long diagonal passes through the center (diameter), length = 2a.

Absolutely. Draw a circle, then mark six points using the radius as chord — the classic Euclidean construction.

Exact: (3√3/2)a². The calculator uses Math.sqrt(3) with double precision — error less than 1e‑15 relative.

Interior angles sum to 720°; each interior angle is 120°, and three hexagons meet at a point (3 × 120° = 360°), allowing gap‑free tiling.
References: MathWorld: Hexagon; Wikipedia: Hexagon; Hales, T.C. “The Honeycomb Conjecture” (1999); Euclid’s Elements, Book IV, Proposition 15.