Calculate the area, perimeter, apothem, circumradius, and diagonals of a regular hexagon from any single measurement such as side length, area, or across flats.
Hexagon Formula
A regular hexagon has six equal sides and six interior angles of 120 degrees. Every other measurement can be written in terms of the side length, so the calculator first converts whatever value you enter into a side length and then derives the rest.
Area = (3 * sqrt(3) / 2) * s^2
Perimeter = 6 * s
Apothem (inradius) = (sqrt(3) / 2) * s
Circumradius = s
Across flats = sqrt(3) * s
Across corners (long diagonal) = 2 * s
Where:
- s = length of one side of the regular hexagon
- Area = surface enclosed by the hexagon, in square units
- Perimeter = total distance around all six sides
- Apothem = distance from the center to the midpoint of a side, also called the inradius
- Circumradius = distance from the center to a vertex, which equals the side length
- Across flats = distance between two opposite parallel sides, equal to the short diagonal
- Across corners = distance between two opposite vertices, the longest diagonal
You choose what you already know from the solve-for menu. If you enter a side length, the area formula is applied directly. If you enter the area, perimeter, apothem, circumradius, across flats, or a diagonal, that value is rearranged back to a side length first. From the side length the calculator returns the perimeter, area, apothem, circumradius, both diagonals, and the across-flats distance at once. The advanced option adds the area of one of the six equilateral triangles, the inscribed and circumscribed circle areas, and unit conversions for the side.
Regular Hexagon Measurement Ratios
Because every dimension is a fixed multiple of the side length, you can estimate any value by hand once you know the side. The table lists the multiplier to apply to the side length s.
| Measurement | In terms of side s | Approximate multiplier |
|---|---|---|
| Perimeter | 6 s | 6.000 |
| Area | (3√3 / 2) s² | 2.598 × s² |
| Apothem (inradius) | (√3 / 2) s | 0.866 |
| Circumradius | s | 1.000 |
| Across flats (short diagonal) | √3 s | 1.732 |
| Across corners (long diagonal) | 2 s | 2.000 |
The reverse direction is just as useful when you measure a real part. The next table shows how to recover the side length from a measurement you can take with a ruler or caliper.
| If you measured | Side length s equals |
|---|---|
| Perimeter P | P / 6 |
| Area A | √(2A / 3√3) |
| Apothem a | 2a / √3 |
| Across flats F | F / √3 |
| Across corners D | D / 2 |
Example Problems
Example 1. You have a regular hexagon with a side length of 10 cm. The area is (3√3 / 2) times 10 squared, which is 2.598 times 100, or about 259.81 square centimeters. The perimeter is 6 times 10, or 60 cm. The apothem is 0.866 times 10, or about 8.66 cm, and the across-corners distance is 2 times 10, or 20 cm.
Example 2. A hex bolt head measures 18 mm across the flats. The side length is 18 divided by √3, which is about 10.39 mm. From that side length the across-corners distance is 2 times 10.39, or about 20.78 mm, and the area is 2.598 times 10.39 squared, or about 280.59 square millimeters.
Frequently Asked Questions
What is the difference between across flats and across corners?
Across flats is the distance between two opposite parallel sides and equals √3 times the side length. Across corners is the distance between two opposite vertices and equals 2 times the side length, so it is always the larger of the two. The across-flats value also equals the short diagonal of the hexagon.
Does this calculator work for irregular hexagons?
No. The formulas assume a regular hexagon where all six sides and all six angles are equal. An irregular six-sided shape needs its actual side lengths, angles, or vertex coordinates, which this tool does not use.
Why does the circumradius equal the side length?
A regular hexagon is made of six equilateral triangles that meet at the center. Each triangle has the center-to-vertex distance and the side of the hexagon as two of its equal sides, so the circumradius and the side length are always the same value.
