Rhombus Calculator

A rhombus is a flat four-sided shape where all four sides have exactly the same length. Because its opposite sides are parallel it is a special case of a parallelogram, and because all sides are equal it is a special case of both. The most distinctive property of a rhombus is its diagonals: they always cross at right angles and each one bisects the other. This means the two diagonals divide the rhombus into four congruent right-angled triangles, which makes calculating all its measurements straightforward once you know the side length and one interior angle. A square is the special rhombus where every angle is 90 degrees and both diagonals are equal. Rhombuses appear in tessellation patterns, crystal structures, engineering cross-bracing, and decorative tiling. This calculator takes the side length and one interior angle between 0 and 180 degrees exclusive and returns the area, both diagonal lengths, the perpendicular height, the perimeter, and the supplementary angle. All results update instantly as you type. The default example uses side = 5 and angle = 60 degrees, giving area = 21.65, diagonals = 5.00 and 8.66, and perimeter = 20.

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units
deg
21.65
area (square units)
Diagonal d15.00
Diagonal d28.66
Height4.33
Perimeter20.00

Angle must be between 0 and 180 degrees exclusive. At 90 degrees the rhombus becomes a square.

How it works

Given side a and interior angle theta in degrees: area = a^2 * sin(theta). The shorter diagonal d1 = 2a * sin(theta/2) and the longer diagonal d2 = 2a * cos(theta/2). The perpendicular height h = a * sin(theta). Perimeter = 4a. The diagonals are derived from the right triangles formed when the diagonals cross at 90 degrees: each triangle has hypotenuse a, so the half-diagonals are a * sin(theta/2) and a * cos(theta/2).

Worked example

For side = 5 and angle = 60 degrees: area = 5^2 * sin(60) = 25 * 0.8660 = 21.65. Shorter diagonal d1 = 2 * 5 * sin(30) = 10 * 0.5 = 5.00. Longer diagonal d2 = 2 * 5 * cos(30) = 10 * 0.8660 = 8.66. Height = 5 * sin(60) = 4.33. Perimeter = 4 * 5 = 20. These match the defaults pre-filled above.

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Frequently asked questions

What is a rhombus?

A rhombus is a parallelogram with all four sides equal in length. Its opposite angles are equal and its adjacent angles add up to 180 degrees. The two diagonals of a rhombus bisect each other at right angles, which is the key property that distinguishes a rhombus from a general parallelogram. A square is a special rhombus where all angles are 90 degrees.

What is the area formula for a rhombus?

The area of a rhombus equals the side squared times sin(theta), where theta is one of the interior angles. It can also be calculated as half the product of the two diagonals: area = d1 times d2 divided by 2. Both formulas give the same result. The sine formula is convenient when you know the side and angle; the diagonal formula is convenient when you have measured both diagonals directly.

How do you find the diagonals of a rhombus?

If the rhombus has side a and one interior angle theta, the two diagonals are d1 = 2a times sin(theta/2) and d2 = 2a times cos(theta/2). These follow from the fact that the diagonals bisect the angles of the rhombus and meet at right angles, forming four right-angled triangles inside the shape.