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CalcMax

Rhombus Calculator

Range: 0 cm – 1,000,000,000 cm

Range: 0 ° – 180 °

Result

21.6506 cm²

Area

Perimeter
20.0000 cm
Diagonal 1
8.6603 cm
Diagonal 2
5.0000 cm
Height
4.3301 cm

A rhombus calculator takes the length of one side and the size of one angle and returns the rest of the figure: the area, the perimeter, both diagonals and the height. A rhombus is a four-sided shape with all four sides the same length and opposite sides parallel — a square that has been pushed over — so one side fixes the perimeter immediately, at four times the side, and the angle then decides everything else. That split is the reason this page asks for two numbers rather than one: the side alone says how long the edges are, and the angle says how far the shape has been leaned, and the two together are exactly enough to pin the figure down. Take a side of 5 centimetres. Stand it up square, at 90 degrees, and it covers 25 square centimetres, which is the most a rhombus of that side can cover. Lean it to 60 degrees and the same four edges enclose 21.6506, about thirteen per cent less, with the diagonals now 8.6603 and 5 and the height 4.3301. Lean it further, to 30 degrees, and the area falls to 12.5 while the perimeter stays at 20 the whole time. That is the fact this page is built to show: leaning a rhombus never changes its perimeter and always costs it area, and the cost reaches everything — the two diagonals move in opposite directions as the shape leans, one growing while the other shrinks, and the height falls away towards zero. The most useful row in the table below is the third one. A side of 5 at 73.74 degrees gives diagonals of exactly 8 and 6, and an area of 24 — which is the same rhombus the rhombus area calculator works from when you give it the diagonals 6 and 8. Half of 6 is 3 and half of 8 is 4, and a right-angled triangle with legs of 3 and 4 has a hypotenuse of 5, which is where the side came from; the angle 73.74 is twice the 36.87 degrees that triangle carries. Enter either pair and the answer is 24, and having both on one screen is the clearest check that the two formulas describe one figure. The angle box takes degrees, radians or gradians, and any of the four angles will do, because the rhombus has two angles of one size and two of the other, and the sine of an angle equals the sine of its supplement — 60 and 120 give the same area. They are the same rhombus, in fact: the same area, perimeter and height, with the two diagonals swapping places, since which one you call the first depends on which corner you measured the angle at. An angle of 0 or 180 is accepted and gives an area of zero, which is the honest reading of a shape that has been flattened completely; a negative angle, or one above 180, is reported as an error instead. If you have the diagonals rather than a side and an angle, the rhombus area calculator takes them directly; if you want only the area from this page's two inputs, that same tool also takes the side-and-angle route.

Rhombuses by side and angle, with the four readings each one gives

Side (cm)Angle (°)Diagonal 1 (cm)Diagonal 2 (cm)Perimeter (cm)Area (cm²)
5608.660352021.6506
5907.07117.07112025
573.74862024
4307.72742.0706168
6120610.39232431.1769
104518.47767.65374070.7107
1101.99240.174340.1736
0600000

Eight rhombuses, and every row shows the same rule at a different lean. The first row is the one the page loads with, and the third is the row to check against another tool: a side of 5 at 73.74 degrees has diagonals of exactly 8 and 6 and an area of 24, which is the same rhombus the rhombus area calculator builds when you give it those two diagonals. The perimeter column never moves, at 20 for every row with a side of 5 and 16 for the row with a side of 4, because no amount of leaning touches the edges — that column and the area column side by side are the clearest statement of what leaning costs. Watch the two diagonal columns as the shape leans. The first row, a side of 5 at 60 degrees, has its longer diagonal in the first column, 8.6603 against 5; the fifth row, a side of 6 at 120 degrees, has the longer one in the second column, 10.3923 against 6. Those two rows are not the same rhombus, since the sides differ, but between them they show the supplement rule: the sine of 120 degrees equals the sine of 60, so leaning a rhombus from an acute angle to its supplement keeps the same two diagonal lengths and merely trades which column each one lands in. The second row is the right angle, the single lean at which the two columns meet, both at 7.0711. The fourth row leans hardest, a side of 4 at 30 degrees, with diagonals of 7.7274 and 2.0706 — the longer one nearly twice the side and the shorter one just over half of it. The last row has a side of 0 and gives zeros throughout — a rhombus with no size, answered rather than refused. Every cell is recomputed from its row when the page is built, in centimetres, degrees and square centimetres.

Formula

A = s² × sin(θ) P = 4s d₁ = 2s × cos(θ ÷ 2) d₂ = 2s × sin(θ ÷ 2) h = s × sin(θ)

s
The length of one side, in centimetres. All four sides are equal, so measuring one is enough, and it fixes the perimeter on its own at 4s. It does not fix the area: a rhombus of side 5 covers anything from 0 up to 25 square centimetres depending on how far it has been leaned, which is why the angle is needed as well
θ
Any one of the four angles, in degrees, radians or gradians. The rhombus has two angles of one size and two of the other, and the two sizes add to 180, so the opposite corners match and adjacent corners supplement. Any of the four works in every formula here, because the sine of an angle and the sine of its supplement are equal — an angle of 60 and an angle of 120 describe the same rhombus: the same area, perimeter and height, with the two diagonals swapping places
A
The area, in square centimetres: the side squared times the sine of the angle. It is at its largest when the angle is a right angle, where the sine is 1 and the rhombus is a square, and it falls to zero as the angle approaches 0 or 180 and the shape flattens into a line. It is printed to four decimals, the same width as the other area pages in this subcategory
P
The perimeter, in centimetres: four times the side. It is the one reading on this page that ignores the angle completely, because leaning the shape moves the corners without changing the edges. A rhombus of side 5 has a perimeter of 20 whether it is a square or nearly flat, which makes the perimeter a poor guide to how much ground a shape covers and a good guide to how much edging it needs
d₁, d₂
The two diagonals, in centimetres: the lines joining opposite corners. They cross at right angles and cut each other in half, and each is twice the side times the cosine or sine of half the angle. As the rhombus leans, one grows and the other shrinks — for a side of 5, both are 7.0711 at 90 degrees, 8.6603 and 5 at 60 degrees, and 9.6593 and 2.5882 at 30 degrees — and their product halves to the area, which is the other route to the same figure
h
The height, in centimetres: the perpendicular distance between one side and the side opposite it, which is the side times the sine of the angle. It is the height you would use with the base-times-height formula, so a rhombus of side 5 at 60 degrees has a height of 4.3301 and an area of 5 × 4.3301 = 21.6506. It falls to zero as the shape flattens, and it equals the side only when the rhombus is a square

Use this page when a rhombus is described by its edge and its corner, which is how the shape is specified in advance: a diamond pane quoted as 5 centimetres at 60 degrees, a tile pattern in a floor, a metal plate cut to a drawing, a kite whose four edges are equal. That description is the one a drawing carries, and it is the one that answers the question a diagonal route cannot: what happens to the area if I lean it further? At a side of 5 the answer runs from 25 square centimetres at a right angle down to 12.5 at 30 degrees and 4.3412 at 10 degrees, and every one of those shapes has a perimeter of 20. Use the rhombus area calculator instead when the diagonals are what you have measured — they are the two distances across a finished shape and they are usually easier to put a tape to than an angle — or when the area is the only reading you want. Use the parallelogram area calculator when the height is a length you already know, since that page takes a base and a perpendicular height directly rather than working the height out from the angle. And use the square calculator when the rhombus turns out to be a square: enter a right angle here and the diagonals come back equal and the height comes back equal to the side, which is what makes the square the largest rhombus of a given side.

Worked examples

  1. A side of 5 centimetres at 60 degrees

    1. Sine of the angle: sin 60° = 0.866025…
    2. Area: 5 × 5 × 0.866025… = 21.650635…, which rounds to 21.6506
    3. Perimeter: 4 × 5 = 20
    4. Half the angle is 30°; cosine 0.866025…, sine 0.5
    5. Longer diagonal: 2 × 5 × 0.866025… = 8.660254…, which rounds to 8.6603
    6. Shorter diagonal: 2 × 5 × 0.5 = 5
    7. Height: 5 × 0.866025… = 4.330127…, which rounds to 4.3301

    The row the page loads with, and the one that shows the whole figure at once. Two of the five readings come straight from the side — the perimeter is 20 and nothing about the angle can change that — and the other three all carry the 0.866 factor of the 60-degree angle, which is the same number that appears in the area of an equilateral triangle. The height of 4.3301 is worth comparing with the side of 5: leaning the square to 60 degrees has cost the shape about thirteen per cent of its height — sin 60° is 0.866, so the loss is 1 − 0.866, a seventh of the height rather than a sixth — and the area fell by the same proportion, from 25 to 21.6506.

  2. A side of 5 centimetres at a right angle

    1. Sine of a right angle: 1
    2. Area: 5 × 5 × 1 = 25
    3. Perimeter: 4 × 5 = 20
    4. Half the angle is 45°; cosine and sine are both 0.707106…
    5. Both diagonals: 2 × 5 × 0.707106… = 7.071067…, which rounds to 7.0711
    6. Height: 5 × 1 = 5

    A square, which is what a rhombus becomes when it has not been leaned at all. Every reading here is a maximum or a coincidence worth knowing: the area of 25 is the largest any rhombus of side 5 can have, the height has grown all the way to the side itself, and the two diagonals have met in the middle at 7.0711 — the square root of fifty, which is the side times the square root of two. The perimeter is still 20, the same as the 60-degree rhombus above, which is the clearest way to see that the perimeter says nothing about how much the shape covers.

  3. A side of 5 centimetres at 73.74 degrees

    1. Half the angle is 36.869897…°, whose cosine is 0.8 and whose sine is 0.6
    2. Longer diagonal: 2 × 5 × 0.8 = 8
    3. Shorter diagonal: 2 × 5 × 0.6 = 6
    4. Area: 5 × 5 × sin 73.739795…° = 25 × 0.96 = 24
    5. Perimeter: 4 × 5 = 20
    6. Height: 5 × 0.96 = 4.8

    The row that ties this page to the rhombus area calculator. Diagonals of 8 and 6 force a side of 5 — the halves are 4 and 3, and a right-angled triangle with those legs has a hypotenuse of 5 — and the angle that goes with that side is 73.74 degrees. So this page and that one meet at 24 square centimetres, reached from a side and an angle here and from two diagonals there. The 0.8 and 0.6 in the steps are the 3-4-5 triangle showing through: any rhombus whose diagonals are in the ratio 4 to 3 has these angles and this side-to-diagonal relation, which is why the numbers come out whole.

  4. A side of 5 centimetres at 0 degrees

    1. Sine of 0°: 0
    2. Area: 5 × 5 × 0 = 0
    3. Perimeter: 4 × 5 = 20
    4. Longer diagonal: 2 × 5 × cos 0° = 2 × 5 × 1 = 10
    5. Shorter diagonal: 2 × 5 × sin 0° = 0
    6. Height: 5 × 0 = 0

    A rhombus flattened completely onto one of its own sides, and the page answers rather than refuses: the area is 0, the height is 0, and the shape has become a doubled line of length 10, which is why the longer diagonal has grown to the full 10 and the shorter has vanished. The one reading that survives is the perimeter, still 20, because the four edges are still there even though the shape enclosing them is not. This is the boundary of what the page accepts: a negative angle is refused as an error, but 0 and 180 are real configurations and are answered as such.

Limitations

This page describes one rhombus from one side and one angle, and it stops there. It does not work backwards: there is no route from an area, a diagonal or a height to the shape that produces it, so the question a fabricator actually asks — what angle gives me 24 square centimetres at a side of 5 — needs the sine worked out by hand, and the answer is 73.74 degrees. All the outputs are in centimetres, square centimetres and degrees regardless of the units chosen for the inputs: the unit dropdowns on the side box convert that length to centimetres, and the angle box changes how the number is read, but neither changes what the answers are printed in. That is a real trap with the angle box, because 60 radians is not 60 degrees and the page will compute the figure for whichever unit is selected rather than complaining that the number is far too large for an angle; if a result looks impossible, the unit selector is the first thing to check. The angle is accepted from 0 up to 180 degrees and outside that range the page reports an error, which quietly assumes you meant the interior angle — an angle given as the exterior one, or as the direction of a side, will produce a figure that is wrong rather than an error. An angle of exactly 0 or 180 gives an area of 0 and a diagonal of 0, which is the honest answer for a flattened shape but is rarely what someone wants, and the page does not warn that the figure it has just described is degenerate. The side is limited to a billion centimetres, which is a limit on exact arithmetic rather than on geometry. Four decimals is a display width shared with the other area pages in this subcategory and not a claim about precision: an angle read to the nearest degree moves the area by around one per cent, far more than the fourth decimal place. The page assumes the shape really is a rhombus, with four equal sides and opposite sides parallel; a quadrilateral that is nearly but not quite one will be answered as though it were, and the rhombus area calculator is the page to use if what you have measured are the diagonals of a shape you are not sure about. Nothing here accounts for waste, kerf or the gaps between tiles, so the area returned is the area of the shape rather than the quantity of material to order.

Frequently asked questions

What can I work out about a rhombus from one side and one angle?
Everything the shape has: the area, the perimeter, both diagonals and the height. One side fixes the perimeter at four times its length, and the angle then fixes the rest, because the rhombus has no other freedom — leaning it is the only thing that changes it once the edges are set. A side of 5 at 60 degrees gives an area of 21.6506 square centimetres, a perimeter of 20, diagonals of 8.6603 and 5, and a height of 4.3301.
Which of the four angles do I enter?
Any of them: the area, the perimeter and the height come out the same whichever one you pick, because every formula here uses the sine of the angle and the sine of an angle equals the sine of its supplement — sin 60 and sin 120 are both 0.866. The two diagonals come out as the same pair of lengths, but swapped between their two rows, since which diagonal is the longer one depends on which corner the angle was measured at. A rhombus has two angles of one size and two of the other, with adjacent corners adding to 180 degrees, so entering 60 and entering 120 describe the same shape. If you know only that one corner is sharp and one is blunt, either number will do.
Does leaning a rhombus change its perimeter?
No. The perimeter is four times the side and the side never changes, so a rhombus of side 5 has a perimeter of 20 whether it is a square or nearly flat. What leaning changes is everything else: the area falls from 25 square centimetres at a right angle to 21.6506 at 60 degrees and 4.3412 at 10 degrees. This is why a perimeter is a poor guide to how much a shape covers and a good guide to how much edging or framing it needs.
How do the two diagonals relate to the side and the angle?
Each is twice the side times the cosine or the sine of half the angle: the longer one is 2s × cos(θ ÷ 2) and the shorter is 2s × sin(θ ÷ 2). At a side of 5 and an angle of 60 degrees that is 8.6603 and 5; at 90 degrees both come out at 7.0711, which is what makes a square's diagonals equal. Their product halved is the area, which is the formula the rhombus area calculator uses when you give it the diagonals instead.
What is the largest area a rhombus with a given side can have?
The square, at the side squared. A rhombus of side 5 covers 25 square centimetres at a right angle, and every other angle gives less: 21.6506 at 60 degrees, 12.5 at 30 and 4.3412 at 10. The reason is in the formula, since the sine of the angle is at most 1 and reaches it at 90 degrees. If you are choosing a corner angle for a tile or a panel and want the most coverage from a given edge length, choose the right angle.
What happens at an angle of 0 or 180 degrees?
The area comes back as 0 and the height as 0, with the perimeter still at 20. Those are honest answers rather than errors: at 0 degrees the shape has been flattened onto one of its own sides and encloses nothing, while the four edges are still there, which is why the longer diagonal has stretched to 10 and the shorter has vanished. Angles outside the range 0 to 180 are reported as errors instead, since the page assumes the number you gave is the interior angle.

References

Related calculators

Rhombus Area CalculatorArea of a rhombus from its diagonals or a side and angle.Parallelogram Area CalculatorArea and perimeter of a parallelogram from its base, height and sloping side.Square CalculatorSide, perimeter or area of a square, plus the diagonal.Triangle Area CalculatorHalf a base times the perpendicular height above it.