A rhombus is a quadrilateral with four equal sides: the diamond on a playing card, the openings in a garden lattice, a square pushed out of shape. It is a special parallelogram, so opposite sides are parallel, and a special kite, so its diagonals cross at right angles. Because all sides match, a rhombus is fixed by its two diagonals or by one side plus one more measurement. This rhombus calculator accepts either and returns the area, side, both diagonals, height, angles and perimeter.
How to use the rhombus calculator
- In I know the, choose Both diagonals, Side and an interior angle, Side and height, Side and one diagonal, or Area and side.
- Fill in the fields that appear: Side (s), Angle α (any interior angle), Diagonal p and Diagonal q, Height (h) (the distance between opposite sides), Diagonal (d) or Area (A).
- Choose the Units, and pick degrees or radians under Angle unit; it sets both the angle you type and the angles in the results.
- Read the area at the top (in Area and side mode the headline is the acute angle). The list gives the side, the long and short diagonals, area, perimeter, height, acute angle α, obtuse angle β and the radius of the inscribed circle. When you enter both diagonals, each keeps the letter you typed it under; otherwise the longer one is called p.
Rhombus formulas
| Quantity | Formula |
|---|---|
| Side from diagonals | s = √((p/2)² + (q/2)²) |
| Diagonals from side and angle | p = 2s cos(α/2), q = 2s sin(α/2) |
| Acute angle from diagonals | α = 2 arctan(q ÷ p) |
| Perimeter | P = 4s |
| Height | h = s sin α |
| Inscribed circle radius | h ÷ 2 |
Worked example: a stained-glass diamond
A leaded window uses diamond-shaped panes measuring 30 cm between the far points and 18 cm across the middle. How much glass and lead does each pane need?
- Choose Both diagonals, enter p = 30 and q = 18, and set the units to centimeters.
- Area: 30 × 18 ÷ 2 = 270 cm² of glass.
- Side: √(15² + 9²) = √306 ≈ 17.49 cm, so the perimeter is about 69.97 cm, roughly 70 cm of lead came per pane.
- Angles: about 61.93° at the narrow points and 118.07° at the wide corners, the settings for the cutting guide.
- Height between opposite sides: 270 ÷ 17.49 ≈ 15.43 cm.
Why the diagonals are perpendicular bisectors
Label the rhombus ABCD. Corner A is the same distance s from B and from D, and so is corner C. Every point equally distant from B and D lies on the perpendicular bisector of BD, so the line through A and C is that bisector. The same argument with B and D shows that BD is the perpendicular bisector of AC.
The result splits the rhombus into four identical right triangles with legs p/2 and q/2 and hypotenuse s. Four triangles of area ½ × p/2 × q/2 add up to pq ÷ 2, and the Pythagorean theorem gives the side. The diagonals also bisect the corner angles, which is where α = 2 arctan(q ÷ p) comes from.
Rhombus, kite and square compared
| Property | Rhombus | Kite | Square |
|---|---|---|---|
| Equal sides | all four | two adjacent pairs | all four |
| Diagonals perpendicular | yes | yes | yes |
| Diagonals bisect each other | yes | only one is bisected | yes |
| Diagonals equal | only if square | not in general | yes |
| Area = pq ÷ 2 | yes | yes | yes, d² ÷ 2 |
A kite with the same 30 cm and 18 cm diagonals as the window pane also covers 270 cm², but its sides come in two different lengths.
How the angle shapes a rhombus
With the side fixed at 10 cm, flattening the angle stretches one diagonal and shrinks the other:
| Acute angle α | Long diagonal p | Short diagonal q | Area |
|---|---|---|---|
| 90° (square) | 14.14 cm | 14.14 cm | 100 cm² |
| 75° | 15.87 cm | 12.18 cm | 96.59 cm² |
| 60° | 17.32 cm | 10 cm | 86.60 cm² |
| 45° | 18.48 cm | 7.65 cm | 70.71 cm² |
| 30° | 19.32 cm | 5.18 cm | 50 cm² |
At 60° the short diagonal equals the side, so the rhombus splits into two equilateral triangles. Three such diamonds form a hexagon, the basis of the “tumbling blocks” quilt pattern.
For a rhombus with right angles, use the square calculator; for unequal neighboring sides, use the parallelogram calculator.
Frequently asked questions
How do I find the area of a rhombus from its diagonals?
Multiply the two diagonals and halve the result: A = pq ÷ 2. Diagonals of 8 cm and 6 cm give 8 × 6 ÷ 2 = 24 cm². The rule works because the diagonals cut the rhombus into four identical right triangles.
How do I find the side of a rhombus from the diagonals?
Half of each diagonal forms a leg of a right triangle whose hypotenuse is the side, so s = √((p/2)² + (q/2)²). Diagonals of 8 and 6 give half-diagonals of 4 and 3 and a side of exactly 5.
What is the difference between a rhombus and a kite?
Both have perpendicular diagonals, so both have area pq ÷ 2. A rhombus has four equal sides and its diagonals bisect each other; a kite has two pairs of equal adjacent sides and only one diagonal is cut in half. Every rhombus is a kite, but most kites are not rhombuses.
Is a square a rhombus?
Yes. A square is a rhombus whose angles are all 90°, which makes its two diagonals equal. Enter a 90° angle with side s and the calculator returns diagonals of s√2 and an area of s².
Why does the calculator show 60° when I entered 120°?
A rhombus has two acute and two obtuse angles, and neighboring angles add to 180°. Entering 120° describes the same shape as entering 60°, so the results list the acute angle α = 60° and the obtuse angle β = 120°.