The area of a regular hexagon is given by the apothem formula A = 2√3·a², where a is the apothem — the perpendicular distance from the center of the hexagon to the midpoint of any one of its six equal sides. This single expression covers every regular hexagon because all six sides and all six apothems of a regular hexagon are equal, so a single measurement fully describes the shape. The constant 2√3 is about 3.4641016, which gives a quick mental estimate of A ≈ 3.4641 × a². The apothem form and the more familiar side-length form A = (3√3/2)·s² are not two competing answers — they are algebraically identical once you use the regular hexagon's defining relationship a = (√3/2)·s. That equivalence is why the Hexagon Area Calculator can give you the apothem, the side length, the perimeter, the circumradius, and the area all in one go from a single side-length input.
The Apothem Formula for Hexagon Area
Every regular polygon — triangle, square, pentagon, hexagon, and beyond — has its area expressed by the same universal apothem rule: A = (1/2) × perimeter × apothem. For a regular hexagon, the perimeter is 6s because every side has the same length s, so the rule becomes A = (1/2) × 6s × a = 3·s·a. The next step depends on which measurement you happen to know.
If you already know the side length, substitute the regular hexagon's apothem identity a = (√3/2)·s to get the well-known A = (3√3/2)·s² ≈ 2.598076·s². If instead you know the apothem directly, solve the same identity for s — s = 2a/√3 — and substitute back:
A = (3√3/2)·s² = (3√3/2)·(2a/√3)² = (3√3/2)·(4a²/3) = 2√3·a².
The 3 and √3 in the numerator and denominator cancel cleanly, leaving 2√3 as the only constant. So the hexagon's apothem formula is the unusually tidy A = 2√3·a², and the side-length formula is the unusually tidy A = (3√3/2)·s². Both constants are irrational, but the algebra collapses without any rounding error, which is why hexagons give exact answers for every input.
Why the Apothem Formula Is Just the Side-Length Formula in Disguise
The reason these two formulas agree is geometric, not algebraic coincidence. Draw lines from the center of a regular hexagon out to each of its six corners, and you cut the shape into six identical equilateral triangles. Each equilateral triangle of side s has area (√3/4)·s², so the six together cover A = 6 × (√3/4)·s² = (3√3/2)·s². That is the side-length form.
Each of those six triangles also has its altitude pointing at the midpoint of one of the hexagon's sides, and that altitude is exactly the apothem a = (√3/2)·s. So the same six triangles, counted through the apothem, give A = 6 × (1/2)·s·a = 3·s·a. Substituting a = (√3/2)·s once more lands on A = 2√3·a². Both formulas describe the same six-triangle dissection; they just count the pieces differently.
This is also why the apothem is sometimes called the inradius — it is the radius of the largest circle that fits inside the hexagon, touching the midpoint of each side. For a regular hexagon the inradius a = (√3/2)·s ≈ 0.8660254·s and the circumradius R = s, so the inscribed circle has a diameter that fills about 86.6% of the circumcircle's diameter, and a hexagon is almost as round as a circle without actually being one.
How to Get Hexagon Area with the Hexagon Area Calculator
The Hexagon Area Calculator takes the side length as its single input. If you only have the apothem on hand, convert it to a side first — s = 2a/√3 ≈ 1.1547·a — then enter s in any unit you like. Everything else is computed for you locally in your browser.
- Enter the side length (s) — the length of one edge of the regular hexagon — in any unit. Centimeters, inches, feet, and meters are all fine because the tool does not convert between them.
- The calculator instantly applies A = (3√3/2)·s² and shows the worked formula with the exact value of √3, so the result is accurate to many decimal places.
- Read the hexagon area below, along with the perimeter (6·s), the apothem (√3/2·s), and the circumradius (s). The apothem line in the output is exactly the value you would have used in A = 2√3·a², so you can cross-check the two formulas against each other.
Because the math runs locally, nothing is uploaded, and the same result appears whether you type into the field on a phone or a desktop. The area comes out in whatever square unit matches your side — square centimeters, square inches, square meters — so there is no hidden conversion to undo.
Worked Example: Side Length Equals 2
Take a regular hexagon with s = 2 units. Plugging into the side-length formula gives A = (3√3/2)·4 = 6√3 ≈ 10.3923 square units. Reading from the same shape's apothem a = √3 ≈ 1.7320508, the apothem formula gives A = 2√3·(√3)² = 2√3·3 = 6√3 ≈ 10.3923, the same answer. The two routes agree exactly because they describe the same six equilateral triangles.
For the same hexagon, the perimeter is 6·2 = 12 units, the apothem is √3 ≈ 1.7320508 units, and the circumradius — the distance from the center to any vertex — is 2 units, equal to the side. That last identity R = s is unique to the regular hexagon and is the reason a hexagon fits snugly around a circle of radius equal to its side.
Other Measurements the Tool Returns
The calculator outputs four values from your single side-length input. They are summarized in the table below, along with the formula form each one depends on, so you can pick the right column for whatever measurement you already have on hand.
| Quantity | Symbol | Formula in terms of s | Formula in terms of a |
|---|---|---|---|
| Side length | s | s | 2a/√3 ≈ 1.1547·a |
| Apothem (inradius) | a | (√3/2)·s ≈ 0.8660254·s | a |
| Perimeter | P | 6·s | 4√3·a |
| Circumradius | R | s | 2a/√3 |
| Area | A | (3√3/2)·s² ≈ 2.598076·s² | 2√3·a² ≈ 3.464102·a² |
The two columns express the same shape in different variables. If your measurement comes from a wrench size across the flats, that width is twice the apothem, so use the right column. If your measurement comes from a ruler placed edge to edge, use the left column.
Where Hexagon Area Calculations Show Up
Hexagons are not just textbook shapes — they appear anywhere a six-sided outline needs to be measured. Hex nuts and bolt heads are the most common everyday example: their width across the flats is 2a, so the wrench size you reach for equals twice the apothem, while the reach of the nut from corner to corner across the flats equals the side s. Knowing the area tells you how much material sits in the head and how much load it can spread.
Honeycomb, hexagonal floor tiles, board-game maps, and chicken-wire mesh all exploit the fact that regular hexagons tile a flat plane with the least perimeter per unit area of any regular polygon. That is why bees build their honey stores as hexagons and why floor tilers choose hex tiles over squares when they want to cover a region with the least grout line. Graphene, the single-atom-thick form of carbon, and the cross-section of a typical pencil lead, both have hexagonal lattices, and their unit-cell area follows the same 2√3·a² rule with a being half the carbon-carbon bond length projected onto the lattice plane.
Outside of geometry, the apothem-based area formula is also handy for students in trigonometry or competition math, where it is often easier to be given the inradius and asked for the area than to be given the side. Engineers and machinists use it when a part spec lists only the across-flats dimension, and quilters and pattern designers use it when laying out hex patches whose spacing is given center-to-edge rather than edge-to-edge.
The Hexagon Area Calculator keeps all of these cases in one place: enter s once and read the apothem, perimeter, circumradius, and area together, with the exact value of √3 preserved so the result does not drift as you change inputs.
For a deeper look, see How Wall Area Translates Into the Paint Tins You Need.
For a deeper look, see Parallelogram Area With Vectors: Cross Product Method.