Hexagon Area Calculator (regular)

A regular hexagon has six equal sides and six 120° angles. It can be split into six identical equilateral triangles, each with side equal to the hexagon’s side.

Area formula:

A = (3√3 / 2) × s² ≈ 2.598 × s²

A 10 cm regular hexagon has area 259.81 cm². The factor 2.598 is roughly 6 × the equilateral-triangle factor 0.4330, which makes sense: six equilateral triangles, each (√3/4) × s², totals 6 × (√3/4) × s² = (3√3/2) × s².

Why hexagons appear everywhere in nature:

The hexagon is the most efficient way to tile a plane with equal-area cells while minimizing total edge length. Honeybees build hexagonal honeycomb because it uses the least wax for the most storage. Cooling lava forms hexagonal columns (the Giant’s Causeway in Northern Ireland). Snowflakes have six-fold symmetry from water’s hydrogen-bond geometry.

In manufacturing and construction:

• Hex bolts and nuts. A hex head lets the wrench re-seat every 60°, so six positions per turn against four on a square head. Half again as many bites in a tight corner, which is the whole reason the square head died out. Standard sizes are stamped as “across flats” (twice the apothem).
• Hex floor tile. Popular in bathrooms and entryways. A 6-in side tile covers 93.5 sq in.
• Asphalt pavers. Many decorative paver patterns use hexagons because they tessellate cleanly.
• Chicken wire and hardware cloth. Hexagonal openings give strength with minimum material.
• Soccer ball panels. Truncated icosahedron pattern: 20 hexagons, 12 pentagons.
• Carbon graphene sheets. A 2D layer of hexagonally-arranged carbon atoms, the basis of one of the strongest materials known.

Worked example: hex floor tile

You’re tiling a 100 sq ft bathroom with 6-inch hex tile. Area per tile = 2.598 × 6² = 93.5 sq in = 0.65 sq ft. Tiles needed: 100 / 0.65 = 154 tiles. Add 10% waste for cuts: 170 tiles.

Other measurements from the same side s:

• Apothem (inradius, centre to mid-side): r = (√3 / 2) × s ≈ 0.866 × s
• Long diagonal (vertex to opposite vertex, through the centre): d_long = 2s
• Short diagonal (vertex to non-adjacent vertex): d_short = s × √3 ≈ 1.732 × s
• Perimeter: P = 6s

Note that the short diagonal and the across-flats distance are the same number. Both come to s√3, because twice the apothem is 2 × (√3/2)s. That coincidence is unique to the hexagon, and it is why one measurement serves for both.

Bolt-head convention. A bolt sized “M10 hex” is quoting thread diameter. The wrench size, measured across the flats of the head, is a separate and much larger number, because the head has to be big enough to grip. An M10 bolt commonly takes a 17 mm wrench under the older DIN standard, and 16 mm under ISO 4014, so check before you buy the socket. Either way the figure is the across-flats distance, s√3.

Sanity-check. The hexagon fits inside its circumscribed circle of radius R = s. That circle has area πs² ≈ 3.14s². Our hexagon area 2.598s² fills about 83% of the circle, against 76% for a pentagon and 64% for a square. More sides means closer to circular.