Calculate everything about a regular hexagon from a single measurement. Choose which value you know — the side length, area, perimeter, a diagonal, or a radius — enter it, and the calculator returns all the other properties instantly.
The hexagon calculator works out every property of a regular hexagon from just one known measurement. Enter the side length and it returns the area, perimeter, both diagonals, the circumradius, and the apothem — or work the other way and enter the area, a diagonal, or a radius to recover the side and everything else. Because a regular hexagon is completely defined by a single length, one number is all it takes to unlock the whole shape. This tool is built for students, teachers, engineers, designers, and hobbyists who need fast, accurate hexagon math without juggling formulas by hand.
Below you will find the complete set of hexagon formulas, worked examples you can reproduce, an explanation of why the hexagon is such a special shape, and answers to the questions people ask most. Everything on this page refers to the regular hexagon — the familiar symmetric six-sided figure with all sides and angles equal.
A hexagon is any polygon with six sides and six angles. A regular hexagon is the special case where all six sides are the same length and all six interior angles are equal. Each of those interior angles measures exactly 120°, and the six angles sum to 720°. A regular hexagon has a high degree of symmetry: six lines of reflective symmetry and six-fold rotational symmetry. One of its most elegant features is that it can be divided into six identical equilateral triangles meeting at the center — a fact that explains almost every formula the calculator uses. Regular hexagons appear throughout nature and design, from honeycomb cells and snowflakes to bolt heads and floor tiles, because they combine efficiency, strength, and the ability to tile a plane with no gaps.
Let a be the side length of a regular hexagon. Every other property follows from it:
| Property | Formula |
|---|---|
| Perimeter | P = 6a |
| Area | A = (3√3 / 2) × a² |
| Long diagonal | D = 2a |
| Short diagonal | d = √3 × a |
| Circumradius (R) | R = a |
| Apothem / inradius (r) | r = (√3 / 2) × a |
A few of these relationships are strikingly simple. The circumradius equals the side length (R = a), which is another way of saying the six corner points sit exactly one side-length from the center. The long diagonal is twice the side (D = 2a), because it passes straight through the center from one vertex to the opposite one. The apothem — the distance from the center to the middle of a side, also called the inradius — is r = (√3/2) a ≈ 0.866 a.
The key to understanding a regular hexagon is to split it into six equilateral triangles that all meet at the center. Each triangle has a side equal to the hexagon's side a, so its own three sides are all a — which is exactly why the circumradius (the distance from center to corner) equals a. The area of one equilateral triangle of side a is (√3/4) a², and six of them give the hexagon area A = 6 × (√3/4) a² = (3√3/2) a². The apothem is just the height of one of those triangles, (√3/2) a. Seeing the hexagon as six triangles turns a list of formulas into a single, memorable picture.
Take the simplest case, a regular hexagon with side length a = 1. Applying the formulas:
This is the calculator's default, so you can press Calculate to confirm it, then enter your own value. Because the shape scales uniformly, doubling the side to a = 2 doubles every length and quadruples the area (to about 10.39), since area grows with the square of the side.
The calculator is fully bidirectional: you do not have to start from the side length. If you know any single property, it first recovers the side a and then computes everything else. The inverse relationships are:
So if you measure the distance across the flats of a hex nut (that is the short diagonal, or twice the apothem) you can immediately find its side length, area, and the corner-to-corner distance — useful for choosing the right wrench or laying out a pattern.
A regular hexagon has two different diagonal lengths, and telling them apart matters. The long diagonal connects two opposite vertices and passes through the center; it is the greatest width of the hexagon, measured corner to corner, and equals 2a. The short diagonal connects two vertices with one vertex skipped between them; it does not pass through the center and equals √3 a ≈ 1.732 a. In practical terms, the long diagonal is the "across the corners" measurement and the short diagonal is the "across the flats" measurement of the hexagon. That across-the-flats distance is exactly twice the apothem, which is why hex nuts and bolts are often specified by it.
Two circles are naturally associated with a regular hexagon. The circumscribed circle passes through all six corners, and its radius — the circumradius R — equals the side length a. The inscribed circle touches the middle of each side, and its radius is the apothem r = (√3/2) a. The apothem is the perpendicular distance from the center to a side, and it plays a starring role in the general polygon area formula, Area = ½ × perimeter × apothem. Plugging in a hexagon's perimeter 6a and apothem (√3/2)a gives ½ × 6a × (√3/2)a = (3√3/2)a², matching the area formula exactly — a nice cross-check.
The regular hexagon is one of only three regular polygons that can tile a flat surface with no gaps or overlaps, the others being the equilateral triangle and the square. Among these three, the hexagon encloses the most area for the least perimeter, which means honeybees can build a honeycomb that stores the maximum amount of honey using the minimum amount of wax. This "honeycomb conjecture" — that hexagonal tiling is the most efficient way to divide a surface into equal cells — was proven mathematically in 1999. The same efficiency shows up in basalt columns like the Giant's Causeway, in the compound eyes of insects, in snowflake symmetry, and in engineered structures from aircraft floors to graphene, where a hexagonal lattice of carbon atoms gives extraordinary strength for its weight.
Hexagon calculations come up in more places than you might expect. Tilers and flooring installers use them to plan hexagonal tile patterns and estimate how many tiles cover an area. Machinists and DIYers rely on the across-the-flats and across-the-corners measurements when working with nuts, bolts, and hex keys. Game designers and quilters lay out hexagonal grids that need precise spacing. Engineers use hexagonal cross-sections and honeycomb cores because they are light yet strong. Even gardeners and landscapers use hexagonal pavers for efficient, attractive layouts. In every case, knowing one measurement and being able to derive the rest saves time and prevents costly mistakes.
The angles of a regular hexagon follow directly from the rules that govern every polygon. The sum of the interior angles of any polygon with n sides is (n − 2) × 180°; for a hexagon that is (6 − 2) × 180° = 720°. Since all six angles are equal in a regular hexagon, each one measures 720° ÷ 6 = 120°. The exterior angles — the turn you make at each corner as you walk around the shape — always add up to 360° for any convex polygon, so each exterior angle of a regular hexagon is 360° ÷ 6 = 60°. That 120° interior angle is the secret behind the hexagon's ability to tile a plane: three hexagons meet perfectly at every point because 3 × 120° = 360°, leaving no gap and no overlap. It is also why the equilateral triangles inside the hexagon fit so neatly, each contributing its own 60° angles at the center.
It is important to remember that this calculator, and every formula on this page, applies to the regular hexagon — the symmetric figure with six equal sides and six equal 120° angles. An irregular hexagon is any six-sided polygon whose sides or angles are not all equal; its shape is not fixed by a single measurement, so there is no single formula for its area or diagonals. To find the area of an irregular hexagon you generally divide it into triangles, calculate each triangle's area, and add them up, or use coordinate geometry with the shoelace formula if you know the corner coordinates. The clean, elegant relationships — area proportional to the side squared, circumradius equal to the side, long diagonal twice the side — are unique to the regular case. When a problem simply says "hexagon" in a geometry class, it almost always means the regular hexagon, but in real-world surveying or design you should confirm the shape is truly regular before applying these formulas.
One of the most useful things to understand about any shape is how its measurements change when you scale it up or down, and the hexagon is a perfect illustration. All the length measurements — perimeter, diagonals, circumradius, and apothem — scale in direct proportion to the side: double the side and each of them doubles. The area, however, scales with the square of the side, because area is a two-dimensional measurement. Doubling the side from 1 to 2 multiplies the area by four (from about 2.598 to about 10.392), and tripling it multiplies the area by nine. This square relationship is why a modest increase in side length produces a surprisingly large increase in area — a fact that matters when estimating material for tiling or when comparing the capacity of hexagonal containers. Keeping the "lengths scale linearly, area scales as the square" rule in mind lets you sanity-check any result the calculator gives you.
The area is A = (3√3/2) × a², where a is the side length. For a hexagon with side 1, the area is about 2.598 square units.
A hexagon has nine diagonals in total. In a regular hexagon these come in two lengths: the long diagonal (2a, corner to opposite corner) and the short diagonal (√3·a).
The apothem is the distance from the center to the midpoint of a side, equal to (√3/2)·a ≈ 0.866·a. It is also the radius of the inscribed circle.
Each interior angle is 120°, and the six angles add up to 720°. The exterior angles are each 60°.
Yes. For a regular hexagon the circumradius (center to corner) equals the side length, R = a, because the hexagon is made of six equilateral triangles.
Rearrange the area formula: a = √(2A / (3√3)). The calculator does this automatically when you choose "Area" as your known value.
This Hexagon Calculator is provided for general educational purposes and applies to regular hexagons, where all sides and angles are equal. Results are computed from standard geometric formulas. For irregular hexagons, whose sides and angles differ, these formulas do not apply.