Rhombus Area Calculator

A = (d₁ × d₂) ÷ 2 A = s² × sin(θ)

d₁, d₂

The two diagonals, in centimetres: the lines joining opposite corners. In a rhombus they cross at right angles and cut each other in half, which is what makes the first formula work. In a square the two are equal, so entering the same number twice is correct rather than a mistake

s

The length of one side, in centimetres. All four are equal, so one is enough. On its own it does not determine the area — a rhombus of side 5 can have any area from nearly nothing up to 25 — which is why this route needs an angle 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; either one can be used in the formula, because the sine of an angle and the sine of its supplement are the same. That is why the page does not care which of the four you measure

A

The area, in square centimetres: half the product of the diagonals, or the side squared times the sine of an angle. Both expressions give the same number for the same rhombus. It is printed to four decimals, the same width as the rest of this subcategory's area pages

÷ 2

The halving in the diagonal formula. It is there because the two diagonals cut the rhombus into four right-angled triangles and the product of the diagonals counts the rectangle around them twice — the rhombus is half of the rectangle whose sides are its diagonals, which is the neatest way to see where the formula comes from

sin(θ)

The sine of the angle, which is the factor that measures how far the rhombus has been leaned. It is 1 at a right angle, so a square has the largest area of any rhombus with that side; it falls to 0 as the shape flattens, so a rhombus leaned almost flat has almost no area. It peaks at 90 degrees and is symmetric about it, which is why an angle of 60 and an angle of 120 give the same area

Use the diagonal route when the rhombus is defined by its diagonals, which is the usual case for anything laid out on the ground or cut from a sheet: a paving stone, a kite, a decorative panel, a window pane set at an angle. Diagonals are what you measure on a finished rhombus, because they are the two distances across it, and half their product is the area — for the 6 by 8 case, 24 square centimetres. Use the side-and-angle route when the shape is defined by its edges and its corners, which is the usual case for anything specified in advance: a tile quoted as 5 centimetres with a 60-degree angle, a diamond pattern in a floor, a plate cut to a drawing. That route also answers the question the diagonal route cannot, which is how much the area changes when you lean the shape: at a side of 5, a rhombus of 60 degrees covers 21.6506 and one of 90 degrees covers 25, so leaning it that far costs about 13 per cent of the area, and leaning it further costs more. Two specific shapes are worth trying. Enter the same number for both diagonals and you have a square — 10 and 10 gives 50, with a side of 7.0711 — which is the largest area a rhombus of that diagonal can have and a useful reference point. And enter a very small angle and the area heads for zero: a rhombus leaned to 10 degrees still has the same side length and almost no area at all, which is the demonstration that the side on its own is not enough to determine the area. If you are comparing this page with the parallelogram area calculator, that one takes a base and a perpendicular height, which also works for a rhombus — the height is the side times the sine of the angle — but asks you for a distance you would have to compute rather than one you can measure. And if you want the perimeter, it is four times the side, which is the one rhombus figure that does not need an angle at all.

1. Diagonals of 6 and 8

   1. Product of the diagonals: 6 × 8 = 48
   2. Halve it: 48 ÷ 2 = 24

   The input the page loads with, and the rhombus the whole page is built around, because it is the one whose two routes can both be checked by hand. Its diagonals of 6 and 8 force a side of 5 — half of each diagonal is 3 and 4, and the triangle with those legs has a hypotenuse of 5 — and the angle that goes with that side is 73.74 degrees. So the other route, a side of 5 with an angle of 73.74, gives the same 24, and the table below prints that row. It is the clearest demonstration on the page that the two formulas describe one shape rather than two.

2. A side of 5 and an angle of 60 degrees

   1. Sine of the angle: sin 60° = 0.866025…
   2. Side squared: 5 × 5 = 25
   3. Area: 25 × 0.866025… = 21.650635…, which rounds to 21.6506

   The side-and-angle route, and the row that shows what leaning a square costs. A square of side 5 covers 25 square centimetres; leaned to 60 degrees the same four sides enclose 21.6506, which is a little over 13 per cent less. The 0.866 figure is the sine of 60 degrees, which is half the square root of three, and it is the same number that turns up in the area of an equilateral triangle — the two are the same geometry seen from different directions.

3. A side of 5 and a right angle

   1. Sine of the angle: sin 90° = 1
   2. Side squared: 5 × 5 = 25
   3. Area: 25 × 1 = 25

   A square, which is a rhombus whose angles have not been leaned at all. The sine of a right angle is 1, so the second formula collapses to the side squared and gives the largest area a rhombus of side 5 can have. Any other angle gives less: at 45 degrees it would be 17.6777, at 30 degrees 12.5, and at 10 degrees 4.3412. That the square is the maximum is worth knowing before choosing an angle for a panel or a tile, since the shape that covers the most ground with a given edge is the one that has been leaned least.

4. Diagonals of 5 and 12

   1. Product of the diagonals: 5 × 12 = 60
   2. Halve it: 60 ÷ 2 = 30

   A long thin rhombus, and a row that is easy to check against its side: half of 5 is 2.5 and half of 12 is 6, so the side is the hypotenuse of a 2.5 by 6 triangle, which is 6.5. That matches the side column of the table. The area of 30 is a good reminder that the two routes are not interchangeable in convenience — the perimeter of this shape is 26, and the area is 30, which is a coincidence of these numbers and not a rule.

5. Diagonals of 1 and 1

   1. Product of the diagonals: 1 × 1 = 1
   2. Halve it: 1 ÷ 2 = 0.5

   The smallest square rhombus, and the row that shows the halving clearly: two diagonals of one give an area of one half, because the rhombus is half of the unit square whose sides are the diagonals. Its side is 0.7071, which is the square root of two over two — the same relationship as a square's diagonal to its side, read the other way round. Note that entering a zero for both diagonals gives an area of zero rather than a blank: a rhombus with no size is a legitimate answer.

The page computes the area of a rhombus and nothing else: it does not print the perimeter, the height, the diagonals or the angles, and the reason is that those all come from the same two measurements you already have. The perimeter is four times the side, which you can read off your own input when you used the side route. The page needs either both diagonals or a side with an angle, and it will report an error if neither route is complete rather than guessing: a single diagonal, or a side without an angle, does not determine the shape. Filling in more than one route is allowed as long as neither is contradicted — a side entered alongside two diagonals is checked against them — but a rhombus whose diagonals imply one side and whose side box says another is reported as a contradiction rather than silently resolved, and the page does not say which of your numbers is the wrong one. The angle box accepts degrees, radians and gradians, and the page assumes the angle is between 0 and 180; an angle of 0 or 180 gives an area of zero, which is the honest answer for a shape that has been flattened, but an angle in a different unit from the one selected will give a wrong answer rather than an error. The outputs are in square centimetres whatever the length dropdowns are set to, so a side entered in inches comes back as a square centimetre area you have to convert yourself. Four decimals is a display width rather than a claim about precision, and it is generous for measurements taken with a tape. The page does not check that the shape is a rhombus: four equal sides and any angles will do, and a shape that looks like a rhombus but whose sides differ slightly will be answered as though it were one, using whichever route you entered. It also has no way to describe a rhombus by its height, and no way to work backwards from an area to the dimensions that produce it, which is the sort of question a panel cut to a fixed area would actually ask.

What is the formula for the area of a rhombus?

There are two, and either is enough. Half the product of the diagonals: a rhombus 6 across one way and 8 the other covers 24 square centimetres. Or the side squared times the sine of any angle: a rhombus with a side of 5 at 60 degrees covers 21.6506. The first is the one to use when you have measured the diagonals; the second when the shape has been specified by its side and its angle. They give the same answer for the same shape, which the table on this page shows directly.

Why does the diagonal formula have a half in it?

Because the rhombus is half of the rectangle whose sides are its diagonals. The two diagonals cross at right angles and cut each other in half, dividing the rhombus into four right-angled triangles, and if you draw the rectangle that has the diagonals as its sides you will find the rhombus occupies exactly half of it. That rectangle has an area of the two diagonals multiplied together, so the rhombus is half that product.

Is a rhombus the same as a diamond?

It is what the word diamond usually means, yes. A diamond shape, a diamond in a pack of cards, a diamond tile — all are rhombuses, which is the four-sided figure with all four sides equal. The word rhombus is the one used in mathematics because it is precise, and the word diamond is the one used on a building site because it is short, and this page is about the same shape either way.

Can I enter both the diagonals and a side and angle?

Yes, as long as they agree. A side entered alongside two diagonals is a consistency check rather than a contradiction, because the diagonals determine the side and the page can tell whether the numbers belong together. What the page will not do is pick a winner: a rhombus whose diagonals imply a side of 5 and whose side box says 7 is reported as a contradiction rather than silently answered from one of them, since neither number is obviously the trustworthy one.

What is the largest area a rhombus with a given side can have?

The square, with an area of the side squared. A rhombus of side 5 with a right angle covers 25 square centimetres, and every other angle gives less: 21.6506 at 60 degrees, 17.6777 at 45, 12.5 at 30 and 4.3412 at 10. This follows from the formula, since the sine of the angle is at most 1 and reaches it at 90 degrees. So if you are choosing an angle for a tile or a panel and want the most coverage from a given edge, choose the right angle.

How do I find the side from the diagonals?

Halve each diagonal and use the two halves as the legs of a right-angled triangle; the side is the hypotenuse. For diagonals of 6 and 8, the halves are 3 and 4, so the side is 5. For 5 and 12 the halves are 2.5 and 6, giving a side of 6.5. This works because a rhombus's diagonals cross at right angles and bisect each other — which is also the fact the area formula is built on.

• Rhombus — the four-sided figure with equal sides this page computes the area of, including the half-the-product-of-the-diagonals result and the relations between the sides, the diagonals and the angles — Wolfram MathWorld (United States)
• Parallelogram — the wider family a rhombus belongs to, with the base-times-height formula that also applies to a rhombus once the height has been worked out from the side and the angle — Wolfram MathWorld (United States)
• Quadrilateral — the general four-sided figure, and the place to check that the diagonal formula used here is the special case of a rule that holds for every quadrilateral whose diagonals cross at right angles — Wolfram MathWorld (United States)
• 教育部关于印发义务教育课程方案和课程标准（2022 年版）的通知——The fifth item in the annex list of this notice is the Mathematics Curriculum Standards for Compulsory Education (2022 edition); the area of a parallelogram and the properties of a rhombus and a square are part of the compulsory-education mathematics curriculum, and the wording of the standards and the grade-band breakdown are governed by that annex — 中华人民共和国教育部