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Hexagon Calculator (Regular)

Area equals three root three over two s squared

Solution

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Calculate Regular Hexagon Area from Side

Use this form when the side length of a regular hexagon (all six sides equal) is known. The formula comes from decomposing the hexagon into 6 equilateral triangles.

A = (3√3 / 2) s²

Calculate Hexagon Perimeter from Side

Use this form to compute the perimeter when the side length is known — fencing, framing, or trim for hexagonal layouts.

P = 6s

Calculate Hexagon Apothem from Side

Use this form for the apothem — the perpendicular distance from the center of the hexagon to the midpoint of any edge. Common in tiling and inscribed-circle calculations.

a = s√3 / 2

Calculate Hexagon Side from Area

Use this rearrangement when the area is known and you need the side length.

s = √(2A / (3√3))

How It Works

This regular hexagon calculator solves A = (3√3/2)s² for area, P = 6s for perimeter, and a = s√3/2 for the apothem (center-to-edge-midpoint distance). The inverse s = √(2A/(3√3)) recovers the side from the area. For a regular hexagon, the circumradius equals the side length (R = s) — a nice property that doesn't hold for other regular polygons. The long diagonal (vertex to opposite vertex) is 2s, and the short diagonal (edge midpoint to opposite edge midpoint) is s√3.

Example Problem

A regular hexagonal tile has a side length of 4 cm. Compute its area, perimeter, apothem, and diagonals.

  1. Knowns: s = 4 cm
  2. Area: A = (3√3 / 2) · s² = (3√3 / 2) · 16 = 24√3 ≈ 41.569 cm²
  3. Perimeter: P = 6s = 6 · 4 = 24 cm
  4. Apothem (center to edge midpoint): a = s√3/2 = 4 · √3 / 2 = 2√3 ≈ 3.464 cm
  5. Circumradius (center to vertex): R = s = 4 cm (special property of regular hexagons)
  6. Long diagonal: 2s = 8 cm; Short diagonal: s√3 ≈ 6.928 cm

The circumradius equals the side length for a regular hexagon because the hexagon decomposes into 6 equilateral triangles meeting at the center — each triangle has all sides equal to s.

When to Use Each Variable

  • Solve for Area — when the side length is known — hexagonal tiles, honeycomb cells, nuts and bolt heads.
  • Solve for Perimeter — when you need the boundary length — hexagonal fencing, trim, or tile edges.
  • Solve for Apothem — when you need the inscribed-circle radius — tooling, machining, or inscribed-pattern designs.
  • Solve for Side — when the area is known and you need the side length.

Key Concepts

A regular hexagon has six equal sides and six equal angles (each 120°). It tiles the plane perfectly without gaps — the reason honeycomb cells, soccer-ball panels, and Allen-key heads use hexagonal geometry. A regular hexagon decomposes into six equilateral triangles meeting at the center. This makes the circumradius (center to vertex) equal to the side length, a special property that holds only for hexagons among regular polygons. The apothem (center to edge midpoint) is the altitude of one equilateral triangle: s√3/2.

Applications

  • Tiling and flooring: hexagonal tiles cover floors and walls without gaps
  • Mechanical engineering: hex nuts, bolt heads, and Allen-key drives
  • Beekeeping and biology: honeycomb cells are hexagonal because hexagons enclose the most area for the least perimeter when tiled
  • Architecture: hexagonal columns, domes, and decorative patterns

Common Mistakes

  • Using the apothem (s√3/2) as the radius — the apothem is the inscribed-circle radius, not the circumscribed circle. The circumradius is s, not s√3/2.
  • Forgetting that a hexagon has 6 sides — perimeter is 6s, not 4s
  • Confusing the regular hexagon (all sides equal) with irregular hexagons (general 6-gon) — this calculator handles the regular case only
  • Squaring the side without computing the (3√3/2) factor — the full coefficient is approximately 2.598, not just π or some other transcendental

Frequently Asked Questions

How do you calculate the area of a regular hexagon?

Use A = (3√3 / 2) · s², where s is the side length. For s = 4 cm, A = 24√3 ≈ 41.57 cm².

What is the formula for the perimeter of a hexagon?

For a regular hexagon, P = 6s (six equal sides times the side length). For s = 4 cm, P = 24 cm.

What is the apothem of a regular hexagon?

The apothem is the perpendicular distance from the center to the midpoint of any side. For a regular hexagon, apothem = s√3/2. For s = 4 cm, apothem = 2√3 ≈ 3.464 cm.

Why does a regular hexagon's circumradius equal its side length?

A regular hexagon decomposes into 6 equilateral triangles meeting at the center. Each triangle has the side length on its base and the circumradius on its two other sides — but all sides of an equilateral triangle are equal, so the circumradius = s. This is unique to hexagons among regular polygons.

How do you find the side of a hexagon given the area?

Rearrange A = (3√3/2)s² to s = √(2A / (3√3)). For A = 41.57 cm², s = √(83.14 / 5.196) = √16 = 4 cm.

What are the diagonals of a regular hexagon?

Two distinct diagonals: the LONG diagonal (vertex to opposite vertex) is 2s, and the SHORT diagonal (edge midpoint to opposite edge midpoint) is s√3. For s = 4, long = 8 and short ≈ 6.928.

Why do hexagons tile the plane perfectly?

Each interior angle of a regular hexagon is 120°. Three hexagons meeting at a vertex sum to 3 × 120° = 360°, exactly filling the plane around that point. Only three regular polygons tile the plane on their own: triangles, squares, and hexagons.

How big is a honeycomb cell?

Real honeycomb cells are about 5–6 mm across (vertex to vertex), corresponding to s ≈ 2.5–3 mm. With s = 3 mm: area ≈ 23.4 mm², apothem ≈ 2.6 mm. The cell is also slightly elongated; this calculator handles the idealized hexagonal cross-section.

Reference: Weisstein, Eric W. "Hexagon." MathWorld — A Wolfram Web Resource. https://mathworld.wolfram.com/Hexagon.html

Worked Examples

Hex Tile

How much surface does a 4 cm hexagonal tile cover?

A hexagonal floor tile has 4 cm sides. Compute its area to count tiles needed for a given floor.

  • Knowns: s = 4 cm
  • Formula: A = (3√3 / 2) s²
  • A = (3√3/2) · 16 = 24√3 ≈ 41.57 cm²

Each tile covers ≈ 41.57 cm² (≈ 0.0042 m²)

A 10 m² floor needs about 10 / 0.0042 ≈ 2,381 tiles, plus 5–10% waste for cuts at the perimeter.

Hex Nut

What is the diameter across the flats of an M10 hex nut?

An M10 hex nut has a wrench size (across-flats) determined by the apothem times 2. For s = 8 mm (typical for M10), find the across-flats distance.

  • Knowns: s = 8 mm (typical M10 hex nut)
  • Formula: apothem a = s√3/2
  • a = 8 · √3 / 2 = 4√3 ≈ 6.928 mm
  • Across-flats = 2a = 8√3 ≈ 13.86 mm

Across-flats ≈ 13.86 mm (M10 nuts are typically spec'd at 14 mm)

Standard wrenches are 14 mm for M10 nuts because the apothem-derived distance rounds up.

Inverse Solve

What side does a 100 cm² hexagonal tile need?

A tile must have exactly 100 cm² area. Find the required side length.

  • Knowns: A = 100 cm²
  • Formula: s = √(2A / (3√3))
  • s = √(200 / (3√3)) = √(200 / 5.196) ≈ √38.49
  • s ≈ 6.204 cm

Side ≈ 6.20 cm

Useful for sizing tiles, hex bolts, or designing hexagonal grids to match a target area per cell.

Regular Hexagon Formulas

A regular hexagon has six equal sides and six 120° interior angles. A single dimension, the side length s, fully determines its geometry.

Area equals three root three over two s squaredPerimeter equals six sApothem equals s root three over twoSide equals the square root of 2A divided by 3 root 3
Regular hexagon with side s and apothem aas

Where:

  • A — area (m², ft², in²)
  • P — perimeter (m, ft, in) = 6s
  • s — side length (all sides equal)
  • a — apothem (center to edge midpoint) = s√3/2
  • R — circumradius (center to vertex) = s (special property of regular hexagons)
  • Long diagonal — vertex to opposite vertex = 2s
  • Short diagonal — edge midpoint to edge midpoint = s√3

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