Hexagon Calculator
A regular hexagon with 10-inch sides has an area of about 259.81 square inches and a perimeter of 60 inches. This hexagon calculator finds the area, perimeter, apothem, circumradius, and both diagonal lengths of a regular hexagon from a single side-length measurement. Enter the length of one side, and every other measurement follows automatically because a regular hexagon is a fixed, predictable shape, with all six sides equal and all six interior angles fixed at 120 degrees. Regular hexagons show up everywhere once you start looking for them, including honeycomb cells, hex bolt and nut heads, hexagonal floor tile, and interlocking paver patterns for patios and walkways. This tool is built for that kind of practical measurement, useful for ordering hex tile by the square foot, checking the wrench clearance across a bolt head, or working out how much trim wraps around a hexagonal planter or gazebo base.
Quick answer
Area is calculated as (3 times the square root of 3, divided by 2) times the side length squared, which measures the flat space enclosed inside the hexagon.
What this tells you
- •Area is calculated as (3 times the square root of 3, divided by 2) times the side length squared, which measures the flat space enclosed inside the hexagon.
- •Perimeter is 6 times the side length, since a regular hexagon has six equal sides.
- •The apothem is the distance from the center to the midpoint of any side, calculated as (the square root of 3, divided by 2) times the side length.
- •The circumradius is the distance from the center to any vertex, and for a regular hexagon it always equals the side length exactly.
- •The long diagonal runs through the center from one vertex to the opposite vertex and equals 2 times the side length.
- •The short diagonal connects two vertices that are two apart and equals the square root of 3 times the side length.
- •All results are rounded to 4 decimal places for consistency, though the calculator uses full floating-point precision internally.
How to Use
- 1Measure or note the length of one side of the hexagon. Since every side of a regular hexagon is equal, only one measurement is needed.
- 2Enter that value into the Side length field. Decimals such as 4.5 or 12.75 are accepted for real-world measurements.
- 3Click Calculate to get the area, perimeter, apothem, circumradius, and both diagonal lengths together.
- 4Match the result to the task at hand. Use area for material coverage such as tile or paint, perimeter for trim or edging, and the apothem or circumradius for fitting the hexagon into a circular space or checking a bolt head across the flats or across the corners.
- 5Enter a new side length anytime to recalculate. There is no need to reload the page or clear the previous result first.
How It Works
Formula
Area = (3 x sqrt(3) / 2) x s^2. Perimeter = 6 x s. Apothem = (sqrt(3) / 2) x s. Circumradius = s. Long diagonal = 2 x s. Short diagonal = sqrt(3) x s.In every formula, s stands for the length of one side. A regular hexagon splits cleanly into 6 equilateral triangles that meet at the center point, and each triangle has a side length equal to s. Since the area of one equilateral triangle is (sqrt(3) / 4) x s^2, six of them add up to 6 x (sqrt(3) / 4) x s^2, which reduces to (3 x sqrt(3) / 2) x s^2, the area formula used here. Perimeter is simple addition, since six equal sides add up to 6 x s. The apothem is the height of one of those equilateral triangles measured from its base to its top point, giving (sqrt(3) / 2) x s. The circumradius is one full side of a triangle reaching from the center to a vertex, so it equals s directly. For example, a hexagon with a side of 6 has an area of about 93.53, a perimeter of 36, and an apothem of about 5.20.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Area and diagonals of a 10-inch hexagonal paver
Area = (3 x sqrt(3) / 2) x 10 x 10 = 2.598076 x 100 = 259.8076 square inches. Perimeter = 6 x 10 = 60 inches. Because the circumradius always equals the side length in a regular hexagon, this paver's center-to-corner distance is also exactly 10 inches, which matters when checking whether it fits inside a 20-inch circular cutout.
Area and perimeter of a 5-inch hex floor tile
Area = (3 x sqrt(3) / 2) x 5 x 5 = 2.598076 x 25 = 64.9519 square inches, close to the coverage of one hexagonal floor tile in a common 5-inch pattern. Perimeter = 6 x 5 = 30 inches, the total edge length that would need grout or trim on an exposed tile.
The unit hexagon: side length of 1
A hexagon with a side of exactly 1 gives an area of 2.5981, a handy reference number. Multiply it by the square of any other side length to get that hexagon's area without recalculating from scratch. For example, a side of 10 gives 2.5981 x 100 = 259.81, matching the paver example above.
Wrench clearance for a 20 mm hex bolt head
A hex bolt head with a 20 mm edge (side) length has an apothem of 17.3205 mm, so the full across-flats distance, the wrench-opening size printed on most bolt specs, is double that, about 34.64 mm. The across-corners distance equals the long diagonal, 40 mm, the wider measurement used when checking clearance in a tight recess.
Regular Hexagon Dimensions by Side Length
Area, perimeter, and apothem for common side lengths, useful as a quick lookup before running your own numbers.
| Side length | Area | Perimeter | Apothem |
|---|---|---|---|
| 1 | 2.5981 | 6 | 0.8660 |
| 2 | 10.3923 | 12 | 1.7321 |
| 5 | 64.9519 | 30 | 4.3301 |
| 10 | 259.8076 | 60 | 8.6603 |
| 20 | 1039.2305 | 120 | 17.3205 |
Circumradius is not listed separately in this table because it always equals the side length shown in the first column.
Why a regular hexagon is six equilateral triangles
Draw a line from the center of a regular hexagon to each of its six vertices, and the shape splits into six identical equilateral triangles, each with all three sides equal to the hexagon's side length. That construction is the source of the hexagon's most useful shortcut: since one side of each triangle stretches from the center to a vertex, the circumradius, the distance from the center to any corner, is always exactly equal to the side length. No other regular polygon has that simple a relationship between its side and its center-to-corner distance.
The same triangle split explains the area formula directly. The area of one equilateral triangle with side s is (sqrt(3) / 4) x s^2. Multiply that by 6 triangles and the result is 6 x (sqrt(3) / 4) x s^2, which reduces to (3 x sqrt(3) / 2) x s^2, the exact formula this calculator uses. The apothem is just the height of one of those triangles, measured from the center out to the midpoint of the opposite side.
Regular hexagons are common outside geometry class for the same structural reason bees settled on them: six equal triangles pack together with no wasted space and share load evenly in every direction. That is why honeycomb cells are hexagonal, why hex bolt and nut heads use six flats instead of four or eight, a good balance between wrench grip and material use, and why hexagonal floor tile and interlocking paver patterns tile a flat surface with zero gaps, unlike circles or pentagons.
Common mistakes
- Confusing the apothem with the circumradius. The apothem is the shorter center-to-edge distance, to the midpoint of a side, while the circumradius is the longer center-to-corner distance, to a vertex. Mixing them up throws off any downstream calculation that depends on one or the other.
- Applying these formulas to an irregular hexagon. Every formula on this page assumes all six sides are equal and all six interior angles are 120 degrees. A hexagon with mismatched sides or angles needs a different method, such as splitting it into triangles and measuring each one separately.
- Doubling the wrong measurement for diagonals. Doubling the apothem gives the across-flats spacing, which equals the short diagonal, while doubling the circumradius gives the long diagonal, the across-corners distance. These are two different distances, not interchangeable versions of the same number.
- Forgetting to square the side length in the area formula. Area scales with the square of the side, so a hexagon with sides twice as long has 4 times the area, not twice the area.
- Mixing units mid-calculation, such as measuring the side in inches but expecting the area in square feet. Convert every measurement to a single unit before entering it.
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