Mathematics

Rhombus: Four Equal Sides and Diagonals That Cross at Right Angles

A rhombus has four equal sides and diagonals that cross at right angles. Yes, a square is a rhombus — and here is the area shortcut using the diagonals.

Rhombus

A rhombus is a quadrilateral with four sides of equal length whose opposite sides are parallel and whose diagonals cross at right angles.

Also called: Diamond
Where students meet it: Grade 5 mathematics, when quadrilaterals are sorted by side and angle properties, and again at GCSE in vector proofs about parallelograms.

The short answer
A rhombus is a quadrilateral with four sides of equal length. Its opposite sides are parallel, its opposite angles are equal, and its diagonals bisect each other at right angles. Every square is a rhombus with right angles, and every rhombus is a parallelogram with all four sides the same.

An example

Area = ½ × 6 × 8 = 24 cm²

A rhombus has diagonals of 6 cm and 8 cm. Half of each product gives an area of 24 cm² without any need for a perpendicular height. The same two numbers also fix the side length: the diagonals cut each other in half at right angles, leaving four right-angled triangles with legs 3 cm and 4 cm, so every side of this rhombus is 5 cm.

What has to be true before it counts

Only one condition is imposed: all four sides the same length. Everything else on the list below is a consequence of that, not an extra requirement, which is why a rhombus can be drawn leaning heavily to one side and still be a rhombus.

Note what is absent. Nothing here says the angles are 90 degrees, and nothing says the diagonals are the same length. In a typical rhombus one diagonal is noticeably longer than the other.

Diagrams mark a rhombus with a single dash on all four sides. If instead one pair carries a single dash and the other pair a double dash, only opposite sides have been declared equal and the shape is a parallelogram, where the diagonals cannot be assumed perpendicular.

Four sides of equal length
Opposite sides parallel, so opposite angles are equal and adjacent angles add to 180 degrees
Diagonals bisect each other at right angles
Each diagonal bisects the two corner angles it passes through
Two lines of symmetry, both running along the diagonals
Rotational symmetry of order 2

Is a square a rhombus?

Yes, and the answer is not a technicality. A square has four equal sides, which is the whole definition, so every square is a rhombus. What a square adds is four right angles, and that extra condition also forces its diagonals to be equal.

The reverse is false. A rhombus is only a square when its angles happen to be 90 degrees, so a question that gives you a rhombus has not given you right angles and you cannot use Pythagoras on the sides as though it had.

Two routes to the area, and when each is quicker

Because a rhombus is a parallelogram, base times perpendicular height works. If a diagram marks a base and a height at right angles to it, that is the shorter route.

When the diagonals are the ones labelled, use half their product instead. The right-angled crossing splits the rhombus into four congruent right-angled triangles, and adding those four areas gives exactly half of one diagonal multiplied by the other.

Common questions

Is a rhombus the same as a diamond?

Diamond is the everyday word for the shape and rhombus is the mathematical one. They describe the same figure, but a diamond drawn on a playing card usually sits balanced on a point, whereas a rhombus can lean any way it likes. Use rhombus in written work; examiners mark the mathematical term.

What is the difference between a rhombus and a parallelogram?

A parallelogram needs opposite sides equal and parallel. A rhombus needs all four sides equal, which is a stricter demand, so every rhombus is a parallelogram but most parallelograms are not rhombuses. The visible difference shows in the diagonals: they cross at right angles in a rhombus and at some other angle in a general parallelogram.

Are the diagonals of a rhombus equal?

Not usually. They cross at right angles and cut each other exactly in half, but their lengths differ unless the rhombus is a square. Equal diagonals are the property of rectangles and squares, and mixing the two facts up is the commonest error on this shape.

How do you find the side of a rhombus from its diagonals?

Halve both diagonals, then treat them as the two shorter sides of a right-angled triangle. Diagonals of 10 cm and 24 cm give legs of 5 cm and 12 cm, and Pythagoras returns a side of 13 cm. The right angle at the crossing point is what makes this work.

Last updated August 21, 2026