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CalcMax

Hexagon Calculator

Range: 0 cm – 1,000,000,000 cm

Result

93.5307 cm²

Area

Perimeter
36.0000 cm
Long diagonal
12.0000 cm
Short diagonal
10.3923 cm
Apothem
5.1962 cm
Interior angle
120 °

A hexagon calculator takes the side length of a regular hexagon and returns six measurements of it: the area, the perimeter, the two diagonals, the apothem and the interior angle. A regular hexagon is the shape of a honeycomb cell, a nut, a bolt head and a floor tile, and the reason it turns up so often is that it is the most efficient way to divide a surface into equal cells with no gaps — which is why bees found it before anyone proved it. Its other virtue is arithmetic: a regular hexagon is made of six equilateral triangles joined at a point, and almost everything about it follows from that one fact. The perimeter is six sides. The long diagonal, the one right across through the centre from corner to opposite corner, is two sides, because it is two of those triangles laid end to end. The short diagonal, the one that skips a single corner, is the height of two of those triangles back to back, which is the square root of three times the side. The area is six equilateral triangles, each of which is a quarter of the square root of three times the side squared, and six of those comes to three times the square root of three over two, times the side squared. And the apothem — the distance from the centre straight out to the middle of a side, which is the radius of the circle the hexagon sits inside — is half the short diagonal, or the square root of three over two times the side. Six outputs, one input. That single input is worth pausing over, because it is what makes this page different from most of the ones around it. A rectangle needs two numbers, a triangle needs three, and a general quadrilateral needs more than that; a regular hexagon needs only its side, because the word regular has already fixed every angle at 120 degrees and every side at the same length. Change the side and the whole shape scales with it — double the side and the perimeter doubles, the diagonals double, but the area goes up by four, since area always goes with the square of length and this page is no exception. The interior angle is the one figure that does not move at all when the side does: it is fixed at 120 degrees by the bare fact that there are six sides, which is worth seeing on the table below, where the same number runs down the last column of every row. The five size figures are printed to four decimal places because three of them carry the square root of three and almost never come out exact, while the perimeter and the long diagonal are exact whenever the side is; the angle is a whole number and is printed as one.

Regular hexagons from a side of zero to a side of ten

Side (cm)Area (cm²)Perimeter (cm)Long diagonal (cm)Short diagonal (cm)Apothem (cm)Interior angle (°)
693.5307361210.39235.1962120
12.5981621.73210.866120
210.39231243.46411.7321120
323.38271865.19622.5981120
564.951930108.66034.3301120
7.5146.1418451512.99046.4952120
10259.8076602017.32058.6603120
000000120

Eight hexagons, and the first row is the one the page loads with. Read down the first two columns to see the one relation that catches people out: going from a side of 5 to a side of 10 doubles the perimeter, the diagonals and the apothem, but the area goes from 64.9519 to 259.8076 — four times, not twice, because area goes with the square of length. The last column is the same 120 on every row, and that is the property rather than a copy-and-paste error: the interior angle of a regular hexagon is fixed by having six sides, so it does not move when the size does, and it is the one figure on this page that survives the side going to zero while all five of the others collapse to nothing. The second row is the unit hexagon, where the three square-root-of-three outputs are at their plainest: 1.7321 for the short diagonal and 0.866 for the apothem, which are √3 and √3 over two. The third row is where the first coincidence sits: a hexagon of side 2 has an area of 10.3923, and the first row's hexagon of side 6 has a short diagonal of the same 10.3923. The seventh row, a side of 10, carries the second: its apothem is 8.6603, which is the short diagonal of a hexagon of side 5 — both of them are 5√3 reached from different directions. Two quantities sharing four printed digits on the same table is exactly the sort of thing that looks like a copy-and-paste error, and both are real. The last row is a side of zero, where the five size columns are zero and the answer is real rather than missing — and where the angle column, for the reason above, is still 120. Every number here is recomputed from its side when the page is built, in centimetres and degrees, and the decimal widths are the same ones the results panel uses.

Formula

A = (3√3 ÷ 2) × s² P = 6s d_long = 2s d_short = √3 × s apothem = (√3 ÷ 2) × s interior angle = 120°

Side
The length of any one of the six sides, in centimetres. There is only this one input because a regular hexagon has all six sides equal and all six angles fixed at 120 degrees, so this number alone determines the entire shape
Area
The space inside the hexagon, in square centimetres, which is three times the square root of three over two, times the side squared. The six equilateral triangles it is made of are where that constant comes from rather than a formula to memorise
Perimeter
The distance all the way around, in centimetres: six times the side, since all six are the same. This is the one output that is always exact whenever the side is, which makes it the one to check the others against
Long diagonal
The distance from one corner straight through the centre to the opposite corner, in centimetres. It is exactly twice the side, which makes it the diameter of the circle drawn around the hexagon, and it is also the widest measurement you can take across the shape
Short diagonal
The distance between two corners with exactly one corner between them, in centimetres, which is the square root of three times the side. It is a little under twice the apothem and a little over one and a half times the side, and it divides the hexagon into a rectangle and two triangles
Apothem
The distance from the centre straight out to the midpoint of a side, in centimetres, which is the square root of three over two times the side. It is the radius of the circle drawn inside the hexagon, and it is exactly half the short diagonal — the two are the same measurement seen from the centre and from a corner
√3
The square root of three, about 1.7320508. It appears in the area, the short diagonal and the apothem, and it is the height of an equilateral triangle of side one — the number that turns the side of a triangle into its height
Interior angle
The angle between two neighbouring sides inside the hexagon, which is 120 degrees for every regular hexagon and therefore does not depend on the side at all. It comes from the rule that any polygon's angles add up to the number of sides minus two, times 180 degrees — six sides, four triangles, 720 degrees shared between six equal corners. It is the one output on this page that a bigger hexagon prints exactly the same
Four decimal places
How wide every output is written. The area, the short diagonal and the apothem almost always carry a √3 and so almost never end; the perimeter and the long diagonal are exact, and printing them to four places is a width rather than a claim. The interior angle is 120 exactly, so it is printed as a whole number with no decimal point at all — the widths are per output rather than one width for the page

The page is for the hexagon as a physical object, and the honeycomb is why it keeps coming up. If you are tiling a surface and want to know how much material a hexagonal tile uses, the area is the figure you want, and it is worth knowing that a hexagon of side one has an area of about 2.6 while a square of side one has an area of 1 — a hexagon covers more ground for the same length of edge, which is the efficiency that makes honeycombs worth copying in packaging and in panels. If you are cutting the tiles out of a sheet, the apothem is what tells you how far apart to space the centres, and the two diagonals tell you how much clearance the finished shape needs — the long diagonal across the widest part, the short one between the flat-to-flat measurements that determine whether a nut will fit a spanner. A bolt head is quoted across the flats, which is twice the apothem, and a socket to fit it is quoted the same way, so the apothem output is the one to reach for rather than the long diagonal. And if you are checking a drawing, the two diagonals have a fixed ratio to each other — the short one is exactly half of the long one times the square root of three — so a computed hexagon whose diagonals do not sit in that ratio has a typo in it rather than a rounding difference.

Worked examples

  1. A side of 6

    1. Perimeter: 6 × 6 = 36
    2. Long diagonal: 2 × 6 = 12
    3. Apothem: (√3 ÷ 2) × 6 = 5.196152…, which rounds to 5.1962
    4. Short diagonal: √3 × 6 = 10.392304…, which rounds to 10.3923
    5. Area: (3√3 ÷ 2) × 36 = 93.530743…, which rounds to 93.5307
    6. Interior angle: (6 − 2) × 180 ÷ 6 = 120 degrees, and it is the same for every regular hexagon

    The input the page loads with. Two of the six outputs are exact — the perimeter at 36 and the long diagonal at 12 — and three carry the square root of three and print four digits. The interior angle is the sixth: 120 degrees, the same figure the table below prints on every one of its eight rows. The check worth doing once: the apothem is exactly half the short diagonal, 5.1962 against 10.3923, and it holds for every side rather than for this one.

  2. A side of 1

    1. Perimeter: 6 × 1 = 6
    2. Long diagonal: 2 × 1 = 2
    3. Short diagonal: √3 × 1 = 1.7320508…, which rounds to 1.7321
    4. Apothem: 0.8660254…, which rounds to 0.866
    5. Area: (3√3 ÷ 2) × 1 = 2.5980762…, which rounds to 2.5981
    6. Interior angle: (6 − 2) × 180 ÷ 6 = 120 degrees, and it is the same for every regular hexagon

    The unit hexagon, and the one to remember the constants from: a regular hexagon of side one has an area of 2.5981, a short diagonal of 1.7321 and an apothem of 0.8660. It also beats a square of side one, which has an area of 1, which is the arithmetic behind the honeycomb. The apothem prints as 0.866 rather than 0.8660 because the fourth decimal is a trailing zero and the printed form is as short as it can be — and the interior angle, at 120, has no decimal part to print at all.

  3. A side of 2

    1. Perimeter: 6 × 2 = 12
    2. Long diagonal: 2 × 2 = 4
    3. Short diagonal: √3 × 2 = 3.4641016…, which rounds to 3.4641
    4. Apothem: 1.7320508…, which rounds to 1.7321
    5. Area: (3√3 ÷ 2) × 4 = 10.392304…, which rounds to 10.3923
    6. Interior angle: (6 − 2) × 180 ÷ 6 = 120 degrees, and it is the same for every regular hexagon

    The row that surprises people: this hexagon's area is 10.3923, and a hexagon of side 6 has a short diagonal of 10.3923. Two different quantities on two different hexagons printing the same four digits is a coincidence of the numbers rather than a repeated calculation, and it is worth knowing about before you spot it in the reference table and assume the worse. The interior angle, by contrast, is the one column with no surprises in it: 120 on every row.

  4. A side of 10

    1. Perimeter: 6 × 10 = 60
    2. Long diagonal: 2 × 10 = 20
    3. Short diagonal: √3 × 10 = 17.320508…, which rounds to 17.3205
    4. Apothem: (√3 ÷ 2) × 10 = 8.660254…, which rounds to 8.6603
    5. Area: (3√3 ÷ 2) × 100 = 259.807621…, which rounds to 259.8076
    6. Interior angle: (6 − 2) × 180 ÷ 6 = 120 degrees, and it is the same for every regular hexagon

    A ten-centimetre hexagon, which is about the size of a small tile. The apothem of 8.6603 is the second of the two coincidences on this page: it is the same four digits as the short diagonal of a hexagon of side 5, since both are 5√3 seen from different places. The figure worth taking from this row is the long diagonal of 20 against the short of 17.32 — the hexagon is wider across its corners than across its flats, by about 15 per cent, which is why a nut and the spanner that fits it are quoted across the flats.

  5. A side of 0

    1. Perimeter: 6 × 0 = 0
    2. Long diagonal: 2 × 0 = 0
    3. Short diagonal: √3 × 0 = 0
    4. Apothem: (√3 ÷ 2) × 0 = 0
    5. Area: (3√3 ÷ 2) × 0 = 0
    6. Interior angle: (6 − 2) × 180 ÷ 6 = 120 degrees, and it is the same for every regular hexagon

    Zero is a legal input rather than an empty box. A hexagon of side zero is a single point and the five size measurements are all zero, so a row of zeros is a real answer rather than a sign that something failed to fill in. The interior angle is the exception and stays at 120 degrees, because it is a property of having six sides rather than of the size — the shape has collapsed, but it has not become a different shape. The page distinguishes a zero from leaving the box blank, which shows no result at all because there is nothing to work from.

Limitations

The page is for regular hexagons only: all six sides the same length and all six angles 120 degrees. A hexagon that is not regular has no single side length and cannot be described by one number, and the page will quietly give you the figures for the regular hexagon with that side rather than telling you that the shape you have in mind is a different one. The interior angle is fixed at 120 degrees for any size and is printed unchanged on every row of the table, including the row where the side is zero; the page gives no exterior angle and no central angle, and no angle at all for a hexagon that is not regular, because there is no single answer to give. The outputs are in centimetres, square centimetres and degrees, whatever unit the dropdown is set to, so a side entered in inches comes back as a centimetre answer you have to convert yourself. Four decimal places is a display width rather than a claim about precision: it is generous for a measured side and the last two digits are not meaningful unless the input was exact. The area grows with the square of the side while the perimeter and the diagonals grow with the side itself, so scaling a hexagon up changes those figures at different rates — the interior angle is the one that does not change at all. The apothem and the short diagonal are printed as two outputs even though one is exactly half the other, because the two are quoted in different trades and the page would rather print both than make you halve one. Nothing here handles a hexagon with a hole through it, a hexagon drawn on a curved surface, or the thickness of a hexagonal tile or nut, which is a third dimension this page has no field for.

Frequently asked questions

Why is one measurement enough for a hexagon?
Because the page is for regular hexagons, where all six sides are equal and all six angles are fixed at 120 degrees. Those two facts leave only the size to decide, and one side length settles it. That is what separates a regular hexagon from a rectangle, which needs two numbers, or a general triangle, which needs three.
What is the apothem, and why is it useful?
It is the distance from the centre of the hexagon straight out to the middle of a side, and it is the radius of the largest circle that fits inside. It is useful because it is how flat-sided objects are measured in practice: a nut and the spanner that fits it are both quoted across the flats, which is twice the apothem, rather than across the corners, which is the long diagonal.
Which diagonal is which?
The long diagonal runs from a corner straight through the centre to the opposite corner and is exactly twice the side. The short diagonal runs between two corners with one corner between them and is the square root of three times the side. The long one is about 15 per cent longer than the short one, and the short one is exactly twice the apothem.
Why do three of the outputs have four decimals and the others none?
Because the widths are set per output rather than for the page. The area, the short diagonal and the apothem all contain the square root of three, which never ends, so their answers almost never do either, and they are printed to four places. The perimeter is six times the side and the long diagonal is twice the side, so both are exact whenever the side is — they are printed to four places as a fixed width, and trailing zeros simply do not show. The interior angle is 120 exactly, so it is printed as a whole number with no decimal point at all.
How does a hexagon compare with a square of the same side?
The hexagon covers more. A regular hexagon of side one has an area of about 2.6 square units against 1 for a square of side one, using the same length of edge for each — but it also has a longer perimeter for that edge, six sides rather than four, so the comparison is only fair when stated as area per unit of edge length. That efficiency is why honeycombs, packaging inserts and radiator panels use hexagonal cells.
Can the side be zero?
Yes, and all five outputs come back as zero, which is the honest description of a hexagon with no size. Zero is treated as a real input rather than as an empty box, so a row of zeros is an answer. A blank box is different: with nothing in it the page shows no result at all, because there is nothing to work from.
Why does the interior angle never change when I change the side?
Because it is a property of the shape rather than of its size. The angles of any polygon add up to the number of sides minus two, times 180 degrees; for six sides that is 720 degrees, and a regular hexagon shares them equally between six corners, which gives 120 each. Nothing in that arithmetic mentions the side length, so a hexagon of side 1 and a hexagon of side 10 both have 120-degree corners — and a hexagon collapsed to a point still reports 120, which is the honest answer for a figure that has lost its size but not its shape.

References

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