How to Find the Area of a Rhombus

A rhombus is a parallelogram with all four sides equal, which means everything true of a parallelogram is true of it. Base times height still works. What a rhombus gains is a second route the general case does not have, and it exists because the diagonals of a rhombus always cross at right angles.

The diagonals are the shortcut

In a general parallelogram the diagonals bisect each other but meet at any old angle. In a rhombus they meet square, every time. That perpendicularity is what makes the diagonal formula possible, and it is why the same trick works on a kite but not on a leaning parallelogram.

It also means the diagonals are usually the easiest thing to measure. On a diamond shaped tile or a garden bed you can run a tape corner to corner in two directions without needing a perpendicular anywhere, and the area comes straight out.

Why halving the product works

There are two ways to see it, and they are the same fact viewed from inside and from outside.

From inside: the diagonals bisect each other, so they cut the rhombus into four right triangles with legs of 5 and 12. Each holds 30, and four of them make 120. Write that in symbols and it is 4 times half of p over 2 times q over 2, which tidies to pq over 2.

From outside: draw the smallest rectangle that contains the rhombus, and its sides are the diagonals themselves, 10 by 24, holding 240. The four corner triangles left outside the rhombus are congruent to the four inside it, so exactly half the rectangle is filled. That is the version worth remembering, because it makes the formula obvious rather than memorable.

A worked check you can repeat on paper

Five routes to one number, and the third and fourth are worth noticing together. The side of 13 falls out of the half diagonals of 5 and 12 by Pythagoras, which is a 5, 12, 13 triangle, so every length in this rhombus is a whole number even though the angles are not. Real rhombuses rarely oblige like this, but it makes the checking easy.

Which formula to reach for

What you have measured	Formula	Worked example
Both diagonals	p q / 2	120
A side and a perpendicular height	b h	120
A side and any angle	a² sin θ	120
One diagonal and the area	other = 2A / p	24

The angle route has a quiet convenience. A rhombus has two different angles, 45.24 and 134.76 degrees here, and it does not matter which one you use, because sin θ and sin of 180 minus θ are equal. Both give 0.7101. That is one of the few places in geometry where picking the wrong one of two things costs you nothing.

Mistakes that catch people out

• Using the half diagonals. The formula wants the full lengths. Feeding in 5 and 12 gives 30, a quarter of the truth, and it is easy to do when a drawing only labels the halves.
• Multiplying two sides together. A rhombus has four equal sides, so 13 times 13 is tempting and gives 169. That is the area only if the rhombus is a square, which it is not unless the diagonals match.
• Forgetting to halve. This gives 240, which is the bounding rectangle rather than the shape inside it.
• Using a diagonal as a height. The height is measured perpendicular from one side to the opposite side, and here it is 9.23, not 10 or 24.
• Assuming a rhombus needs a right angle. It does not. Its diagonals meet at right angles, but its corners are usually nothing of the sort.

Try these

Common questions

What is the formula for the area of a rhombus?

Multiply the two diagonals and halve the answer. Diagonals of 10 and 24 give 120 square units. Base times height and a² sin θ both work as well, since a rhombus is a parallelogram.

Why do you halve the product of the diagonals?

Because the rhombus fills exactly half of the rectangle its diagonals define. A 10 by 24 rectangle holds 240, and the four corners left outside the rhombus match the four triangles inside it.

Is a rhombus a parallelogram?

Yes, one with all four sides the same length. Every parallelogram formula applies to it, and it gains the diagonal shortcut because its diagonals cross at right angles.

How do you find the side from the diagonals?

Halve each diagonal and use Pythagoras. Diagonals of 10 and 24 give halves of 5 and 12, so the side is √(25 + 144), which is 13.

Is a square a rhombus?

Yes. A square is a rhombus whose diagonals happen to be equal. The diagonal formula still works on it: two diagonals of 10 give 50, which matches a side of √50 squared.

Does it matter which angle I use?

No. A rhombus has two angles that add to 180 degrees, and their sines are identical, so a² sin θ gives the same answer either way. Here 45.24 and 134.76 degrees both give a sine of 0.7101.

Related lessons and calculators

• Area of a parallelogram, the family this shape belongs to, where base times height and a² sin θ are set out properly.
• Area of a trapezoid, the other quadrilateral where one formula quietly contains several shapes.
• Missing side of a right triangle, the Pythagoras step that turns half diagonals into a side length.
• Area formulas for every common shape, with a calculator that runs the rhombus from its diagonals.
• Area of 2D shapes calculator, for running these without the arithmetic.
• the National Council of Teachers of Mathematics, for how quadrilaterals are classified and taught.
• Area of a hexagon, the regular polygon that comes apart into six equilateral triangles.
• Base of a parallelogram, worked back from an area, a perimeter, an angle or a pair of diagonals.