How this instrument works
A regular hexagon has six equal sides and six interior angles of exactly 120°, and its area follows from a single side length: A = (3√3 ⁄ 2)s². The formula comes from a construction worth picturing — draw a line from the center to each vertex and the hexagon splits into six identical equilateral triangles, each with area (√3 ⁄ 4)s². Six of those triangles sum to (3√3 ⁄ 2)s² exactly, with no rounding introduced by the split itself.
That construction hides a fact unique to this six-sided case: the segment from the center to a vertex (the circumradius) is exactly equal to the side length, not merely close to it. This happens because the angle at the center of each wedge is 360° ⁄ 6 = 60°, which is precisely the angle that turns an isosceles wedge into an equilateral triangle. A pentagon or a heptagon divides into isosceles wedges too, but the center angle in those cases is never 60°, so their radius and side length stay different numbers.
The 120° corner is also why regular hexagons tile a flat surface without gaps or overlaps — three of them meet at a point (3 × 120° = 360°) and close perfectly. Thomas Hales proved in 1999 that this hexagonal tiling is the most efficient way to divide a plane into equal-area cells using the least total wall length, a result now called the honeycomb conjecture. Bees build in hexagons for the same reason a paving contractor might: it is the cheapest boundary for the area it encloses. At the other extreme, a side length of zero collapses the hexagon to a single point, and both area and perimeter fall to zero.
- Enter your hexagon's edge length into the Side length field — any unit is fine, since area and perimeter come back in that same unit.
- Read Area (regular hexagon) for the enclosed surface, computed as A = (3√3 ⁄ 2)s².
- Read Perimeter for the total boundary length, which is simply six times the side.
- If you are measuring a real object such as a nut, tile, or paving stone, run a straight edge along one flat side rather than across the corners or across the flats.
Worked example — a hexagon with 4 cm sides
A hexagonal paving stone measures 4 cm along each edge, so s = 4. Area (regular hexagon) comes out to (3√3 ⁄ 2) × 4² = 41.569219381653056 cm², which is 24√3 cm², about 41.57 cm² once rounded. Perimeter reads 6 × 4 = 24 cm, the exact length of edging strip needed to trim the stone all the way around.
The six-triangle picture checks the same number a second way: one equilateral triangle with a 4 cm side has area (√3 ⁄ 4) × 4² = 4√3 ≈ 6.928203 cm², and six of them sum to 24√3 ≈ 41.569219 cm², matching Area (regular hexagon) exactly. It also shows why size scales unevenly here — stretch the side to 8 cm and the area rises to roughly 166.28 cm², a fourfold jump, while the perimeter only doubles to 48 cm.
Questions
What is the area formula for a regular hexagon?
A = (3√3 ⁄ 2)s², where s is one side. It comes from splitting the hexagon into six equilateral triangles meeting at the center, each contributing (√3 ⁄ 4)s². For s = 4 that gives 24√3, about 41.57 square units — the same figure this sheet returns in the Area (regular hexagon) field.
Why does a hexagon tile a flat surface with no gaps?
Its interior angle is 120°, and three of those meet at a point to close a full 360° turn with nothing left over. Thomas Hales proved in 1999 that this hexagonal grid is also the most efficient way to divide a plane into equal-area cells with the least total boundary — the honeycomb conjecture, named for the shape bees settle on independently.
Is a hexagon's circumradius really equal to its side length?
Yes, and only for the hexagon among regular polygons. The angle at the center of each wedge is 360° ⁄ 6 = 60°, which turns that wedge into an equilateral triangle rather than a generic isosceles one, so the center-to-vertex distance and the side share the same value exactly.
How is the perimeter of a regular hexagon calculated?
Perimeter is P = 6s, six times the length of one side, since a regular hexagon's six sides are equal by definition. If a hexagon is irregular, that shortcut fails and you would instead measure and add all six edges individually.
Does doubling the side length double the area?
No — area grows with the square of the side, so doubling s roughly quadruples it. A 4 cm side gives about 41.57 cm² of area, while an 8 cm side gives about 166.28 cm², four times as much, even though the perimeter only doubles from 24 cm to 48 cm.
What's the most common measuring mistake with this calculator?
Entering the width across flats instead of the edge length. Hex bolt and nut specifications often list the flat-to-flat distance, which is the apothem doubled, not a side. To convert, divide the flat-to-flat width by √3 to recover the side length s this sheet expects.