Rhombus area by diagonals:
formula and worked example

Using diagonals when sides are not enough

A rhombus is a four-sided shape with all sides equal, a tilted square. Diamond-shaped floor tiles, kite-like panels, rhombic roof sections and the diamond cut on a playing card are all examples. Often you can measure the two crossing diagonals more easily than the height, because a height requires a perpendicular drop and a clear base line.

For a rhombus the diagonals meet at right angles and cut each other in half. That special property is what lets area be written in terms of nothing but the two diagonal lengths.

The formula and its two inputs

Area equals the first diagonal times the second diagonal, divided by two. Call the long one d1 and the short one d2; the order does not matter. The diagonals run from opposite corners, and each one passes through the centre where they cross.

Geometrically, drawing a rectangle of d1 by d2 around the rhombus places each corner on the middle of a rectangle side, and the rhombus occupies exactly half. That is where the divide-by-two comes from.

Equivalent formulas exist for other starting data: side times height, or side squared times the sine of an interior angle. Here the side squared is 74 and the interior angle is about 71.1 degrees, whose sine is 0.946, so 74 x 0.946 gives 70 again; the methods agree when the data are consistent.

Diagonals of 10 and 14

A diamond-shaped panel has diagonals of 10 units and 14 units.

1. 1Multiply the diagonals: 10 x 14 = 140.
2. 2Divide by two: 140 / 2 = 70.
3. 3State the unit: 70 square units.

Confirm it by dividing the rhombus into four right triangles. Each has legs of 5 and 7 (half of each diagonal), so each has area 5 x 7 / 2 = 17.5, and four of them give 70. The side length also drops out as the hypotenuse, root of 25 + 49 = about 8.60.

What the number is for

The area is the total surface of one panel, tile or cut-out. For a floor pattern of diamond tiles it is the covered surface per tile; divide the room area by it to estimate the count and add the usual wastage for cut edges at walls. For a metal diamond plate or a sail section the area leads straight into weight and cost.

Two rhombuses with the same side length can have very different areas. A rhombus with sides of 8.6 and a nearly flat shape is much smaller than a square with sides of 8.6, because area depends on how open the angle at the corners is. Sides alone do not determine the surface; a diagonal or an angle must be known as well.

When the diagonals are equal, the rhombus is a square and the area simplifies to half of the diagonal squared. When one diagonal is much shorter than the other the shape flattens toward a line and the area shrinks accordingly, even though the sides remain equal.

Lengthening a single diagonal

With d1 moving from 10 to 11, the area goes from 70 to 77, exactly 10% more. That proportional response happens because the area is directly proportional to each diagonal separately. If both diagonals are scaled by 10%, the area rises by 21%, so scaling a shape uniformly always follows the square law.

This makes design adjustments easy to price. Lengthening a diamond by one unit along d1 adds d2/2 = 7 square units no matter how long d1 already is.

It is useful to see what lengthening a diagonal does to the shape, not just the area. Stretching d1 from 10 to 11 with d2 fixed keeps the diagonals perpendicular but makes the side longer, about 8.60 going to 8.90, so each edge, and any trim along it, grows too.

Errors specific to the diagonal method

• Forgetting the halving and quoting 140 instead of 70.
• Multiplying the side length by itself, which gives the area of a square with the same side, not the rhombus. For this one that would be about 74, close enough to look right.
• Applying the formula to a parallelogram or a rectangle. Their diagonals are not perpendicular, so half the product is not their area.
• Confusing the diagonals with the sides in a drawing, especially in rotated diamonds where the diagonals look horizontal and vertical.

Scenario: laying diamond flooring tiles

A hallway is tiled with rhombic tiles whose diagonals are 30 cm and 40 cm. Each tile covers 30 x 40 / 2 = 600 square centimetres, and the side of each is 25 cm (a 15-20-25 right triangle). A 6 square metre area equals 60,000 square centimetres, so 100 tiles cover it, plus roughly 10% for cutting at the edges. Ordering by area from the diagonals is quicker than working from sides and angles, and it avoids needing the corner angle at all.

The same figure also gives the shortest route to cost. If each tile costs a fixed amount per square metre, the price per tile is area times that rate, and because area comes from two quick measurements, a whole pattern can be priced from a single tile placed on the floor.