Hexagon Calculator Explained: Every Length the Side Implies

What a regular hexagon actually is

A regular hexagon is a six-sided polygon with all six sides the same length and all six interior angles equal to exactly 120 degrees. That second condition is the one most people forget. A shape can have six equal sides and still be irregular — a squashed hexagonal frame, for instance — if the angles between the sides are not all equal. The hexagon calculator on this page works only on the regular case, because that is the case where a single number (the side length s) defines everything else about the shape.

Regular hexagons show up far more often than people give them credit for. Honeycomb cells. The cross-section of a standard bolt head or nut. Bathroom and kitchen tiles that have to fit flush without gaps. Board-game maps. Allen-key recesses. Carbon rings in organic chemistry. Pencil cross-sections. Each of these relies on the same handful of identities that fall out of one side length, and those identities are what this article unpacks.

The big idea, the one that makes every formula simple, is that a regular hexagon is six congruent equilateral triangles meeting at its centre. Once you see the hexagon that way, every length and area you might want is just a rearrangement of properties you already know about the equilateral triangle.

How the hexagon formulas are derived

The six identities the hexagon calculator returns are perimeter, area, apothem, long diagonal, width across the flats, and circumradius. Five of them follow from the equilateral-triangle decomposition; the sixth is just six copies of one side.

Perimeter

P = 6s

Six equal sides, each of length s. Nothing clever — this one is here so the calculator is honest about what it knows. The only practical wrinkle is unit-handling: if s is in centimetres, P is in centimetres; if s is in feet, P is in feet. The formula is scale-invariant.

Area

A = (3√3 / 2) · s² ≈ 2.598 · s²

Drop in the equilateral-triangle area, which is (√3 / 4) · s², and multiply by six:

A = 6 · (√3 / 4) · s² = (3√3 / 2) · s²

The √3 appears because the height of an equilateral triangle with side s is (√3 / 2) · s. Halving the base and multiplying by that height gives one triangle’s area; six of them gives the hexagon. The factor 2.598 is worth memorising: a hexagon with side s is just under two-and-a-half times s in area. A square with side s is, of course, exactly 1 · s², so a hexagon punches well above a square of the same side length.

Apothem (inradius)

a = (√3 / 2) · s ≈ 0.866 · s

The apothem is the perpendicular distance from the centre of the hexagon to the midpoint of any side. It coincides with the radius of the largest circle that fits inside the hexagon — the inradius. Geometrically it is just the height of one of the six equilateral triangles, which is (√3 / 2) · s. In practical work the apothem is most often used as half of the “width across the flats” (see below), the dimension that matters when you size a hex nut to a wrench or fit a hexagonal tile inside a circle.

Long diagonal (the vertex-to-vertex diameter)

d₁ = 2s

The long diagonal connects two opposite vertices through the centre. Because the circumradius equals the side (proof below), the long diagonal is the diameter of the circumscribed circle, which is 2 · R = 2s. This is the largest straight-line distance you can draw inside a regular hexagon.

Width across the flats (the side-to-side diameter)

w = √3 · s ≈ 1.732 · s

This is the perpendicular distance between two opposite parallel sides. Twice the apothem — 2 · (√3 / 2) · s = √3 · s — it is the dimension stamped on bolts and printed in wrench catalogues. In imperial socket sets you will see this called “AF” (across flats); in metric you will see it as the size in millimetres, e.g. M10 nuts are typically 17 mm AF. The long diagonal is always 2·s, but the across-flats width is about 13% smaller at √3·s, which is why a hex-headed bolt fits a slightly smaller wrench than its vertex-to-vertex diameter would suggest.

Circumradius

R = s

Of all the regular polygons, only the hexagon has the property that the radius of its circumscribed circle equals its side length. The proof is short: connect the centre to two adjacent vertices. Both lines are radii, so they are equal. The angle between them is the central angle, which for a six-sided polygon is 360°/6 = 60°. A triangle with two equal sides and a 60° angle between them is equilateral, so its third side — which is the hexagon’s side — equals the radii. So R = s.

Worked example

Take a regular hexagon with side length s = 10 (the calculator defaults to this value, so plug 10 into the hexagon calculator to follow along).

• Perimeter: P = 6 · 10 = 60 units.
• Area: A = (3√3 / 2) · 100 = 150√3 ≈ 259.808 square units.
• Apothem: a = (√3 / 2) · 10 = 5√3 ≈ 8.660 units.
• Long diagonal: d₁ = 2 · 10 = 20 units.
• Width across flats: w = √3 · 10 ≈ 17.321 units.
• Circumradius: R = 10 units.

Now scale the side to 25 cm and the ratios stay constant: P = 150 cm, A = 150 · 6.25 · √3 ≈ 1,623.8 cm², apothem ≈ 21.65 cm, long diagonal 50 cm, width across flats ≈ 43.30 cm. That last figure is the practical one for tile-layers: a 25 cm regular hex tile sits in a 43.30 cm strip, not a 50 cm strip, so the row spacing on a flat-up orientation is 43.30 cm centre-to-centre, not 50 cm. Getting that wrong by 13% is the classic short-order in hexagonal tile work.

Where the hexagon shows up in real work

Honeycomb and the honeycomb conjecture

Bees build their cells as hexagonal prisms because of a deep result in geometry: the regular hexagon is the most efficient way to tile a plane, meaning it encloses the largest area for the smallest perimeter. The intuition is straightforward — the only regular polygons that tile a flat surface without gaps are triangles, squares, and hexagons, and of those three the hexagon is the closest to a circle while still packing flush. The formal statement is the “honeycomb conjecture”, posed by Pappus of Alexandria in the 4th century and finally proved by Thomas C. Hales in 1999. For bees it means the least wax for the most honey storage. For tile manufacturers it means the same square metre of floor needs the fewest cuts at the edges if you use a hexagonal pattern.

Nuts, bolts, and tools

The hex head is standard on bolts because six flats are the minimum that still let a wrench engage from any of six rotational positions, which is enough to tighten or loosen in cramped quarters. The dimension a wrench grips is the width across the flats — √3 · s — not the vertex-to-vertex distance. So a 17 mm spanner fits an M10 nut not because the nut is 17 mm corner-to-corner (it is closer to 19.6 mm) but because the perpendicular distance across the flats is 17 mm. The same goes for hex-key recesses, where the key’s body width matches the recess’s across-flats dimension.

Tile and floor layout

Hexagonal tiles can be laid in two orientations: flat-up (a pair of parallel sides horizontal) and point-up (a pair of opposite vertices on the vertical axis). The row spacing in flat-up is the across-flats width √3·s; the column spacing is 1.5·s with a half-offset between adjacent columns. Point-up swaps these. The square footage calculator handles the total area to cover; the hexagon calculator handles the geometry of a single tile, and the two together give you both the order quantity and the cut allowance.

Board games and hex maps

Wargames, role-playing systems, and abstract strategy games use hex grids because every cell has six neighbours at the same distance from the centre — unlike a square grid, which has four edge-neighbours at distance 1 and four diagonal neighbours at distance √2. Hex grids make movement and line-of-sight calculations clean. The standard cell size is the across-flats width, because that determines how many cells fit across a fixed-width board.

Organic chemistry

Benzene, the foundational aromatic compound, is drawn as a regular hexagon with alternating double bonds (or, more accurately, as a hexagon with a circle inside to denote delocalised electrons). The carbon atoms sit at the vertices, which is why organic chemists draw molecular skeletons as zig-zags of triangles and hexagons. The same geometry repeats in graphite, graphene, and every aromatic ring in biochemistry.

Hexagons compared with other regular polygons

For the same side length s, here is how a hexagon stacks up against its closest neighbours:

• Triangle (3 sides): area = (√3 / 4) · s² ≈ 0.433 · s².
• Square (4 sides): area = s².
• Pentagon (5 sides): area ≈ 1.720 · s².
• Hexagon (6 sides): area ≈ 2.598 · s².
• Octagon (8 sides): area ≈ 4.828 · s².

Area grows faster than side count, which makes sense — more sides means closer to a circle, and a circle encloses more area than any polygon of the same perimeter. The reason designers stop at hexagons rather than going higher is the tiling property: heptagons, octagons, and beyond do not tile a flat plane on their own. Hexagons are the largest regular polygon that still pieces together without gaps. That single constraint is why hexagons dominate so much of the real engineered world.

Common mistakes

Confusing the two diagonals

The long diagonal (vertex-to-vertex) and the width across the flats (side-to-side) are the two natural “diameters” of a hexagon, and they are not the same length. Long diagonal is 2s; width across flats is √3 · s, smaller by about 13%. Conflating the two will land you the wrong wrench for a bolt or the wrong board-game grid spacing for a printed map.

Applying regular-hexagon formulas to irregular shapes

If any side or angle differs from the others, the shape is no longer regular and the closed-form expressions on this page break. The fallback is to split the polygon into triangles, compute each triangle’s area with the shoelace formula (or with base×height/2), and sum them. The hexagon calculator assumes regularity by design and will give nonsense if you feed it the side of an irregular six-sided shape.

Mixing units

A side in centimetres gives an area in square centimetres, not square metres. A side in feet gives an area in square feet, not square yards. The calculator is unit-agnostic on input, which means the user has to keep track of unit on output. If you need to translate between metric and imperial areas afterwards, use the area converter.

Assuming hexagonal area equals 6 × equilateral side²

It is six equilateral triangles, each of area (√3/4)·s², not six s² squares. The factor matters: 6 · s² would be the area of six separate squares with side s, which is more than double the hexagon’s area. The correct multiplier is roughly 2.598, not 6.

When the calculator is not enough

The hexagon calculator is a single-input tool: feed it a side length, get back the six derived lengths and the area. It does not handle irregular hexagons, hexagonal prisms (3D), tessellation layout (how many tiles cover a floor), or non-Euclidean geometries. For each of those you need a different approach:

• Irregular hexagons: split into triangles and use the shoelace formula. The general formula for a polygon’s area from vertex coordinates (x₁, y₁) ... (xₙ, yₙ) is ½ · |∑_i (x_i · y_i+1 − x_i+1 · y_i)|.
• Hexagonal prisms: volume = hexagon area × height = (3√3 / 2) · s² · h. Surface area = 2 × hexagon area + 6 × (s · h) for the side faces.
• Tile layout: divide the room area (from the square footage calculator) by one hexagon’s area, then add 10–15% for cuts and waste — closer to 15% on rooms with many edges or curves.
• Spherical or curved surfaces: the formulas here are Euclidean and assume a flat plane. A hexagon on a sphere (the panels of a soccer ball, parts of a geodesic dome) follows spherical geometry where interior angles add to more than 720°, and the closed-form area expression differs. For those cases, use spherical-trigonometry tools rather than this calculator.

Frequently asked questions

What does “regular” hexagon mean, and why does it matter for the formulas?A regular hexagon has six sides of equal length and six interior angles of exactly 120 degrees. Every formula above assumes that regularity. Irregular hexagons need a different approach: split them into triangles and add the triangle areas.

Why is the hexagon area formula (3√3/2)·s²?A regular hexagon is exactly six congruent equilateral triangles meeting at its centre. Each triangle of side s has area (√3/4)·s², so the hexagon contains six of those, which is (3√3/2)·s² ≈ 2.598·s².

What is the apothem of a hexagon used for?The apothem is the perpendicular distance from the centre to the midpoint of any side and equals (√3/2)·s. It is the inradius of the largest circle that fits inside the hexagon, and twice the apothem is the width across the flats — the dimension that matters for nuts, bolts, and hex tiles.

Why does the circumradius equal the side length in a hexagon?Joining the centre to two adjacent vertices forms a triangle with two equal sides (both radii) and a 60° central angle. That triangle is equilateral, so the third side — the hexagon’s side — equals the radii: R = s. Only the hexagon among regular polygons has this identity.

What is the difference between “long diagonal” and “width across the flats”?The long diagonal joins two opposite vertices through the centre and equals 2s. The width across the flats is the perpendicular distance between two opposite parallel sides and equals √3 · s ≈ 1.732 · s. The long diagonal is always about 15.5% larger than the across-flats width.

Why are hexagons the most efficient way to tile a flat surface?Of the three regular polygons that tile a plane without gaps — triangles, squares, and hexagons — the hexagon encloses the largest area per unit perimeter. The result is the “honeycomb conjecture”, proved by Thomas C. Hales in 1999. Bees use it to minimise wax; tile-layers use it to minimise edge cuts.

How do I work out the area of an irregular six-sided polygon?Split the polygon into triangles by drawing diagonals from one vertex, compute each triangle’s area, and sum. The shoelace formula is the most general approach when you know the vertex coordinates. The regular-hexagon formulas on this page do not apply.

How does the hexagon area scale if I double the side?Area is proportional to s², so doubling s quadruples the area. The perimeter only doubles, so larger hexagons enclose more area per unit perimeter than smaller ones — a scaling law that underlies why honeycomb cells are sized to fit a bee, not a microbe.

Related calculators

Pair the hexagon calculator with these tools for the rest of the geometry workflow.

• Square footage calculator — rectangular, triangular, and circular floor areas with a cost-per-ft² option, useful for sizing tile orders.
• Slope calculator — slope, line equation, and point-to-point distance, the companion piece for coordinate-geometry work.
• Area converter — convert between m², ft², acres, and hectares, for when the hexagon area lands in the wrong unit for the downstream task.