Rhombus – Definition, Properties, Formulas & Examples

A rhombus is a four-sided flat shape (a quadrilateral) where all four sides are equal in length and the opposite sides are parallel. The diagonals of a rhombus cross each other at right angles and cut each other in half.

Every rhombus is a parallelogram, every rhombus is a kite, and every square is a rhombus. This article covers the definition of a rhombus, its seven key properties, three area formulas, the diagonal–side relationship, how to verify a quadrilateral is a rhombus, and worked examples.

What Is a Rhombus?

A rhombus is an equilateral quadrilateral — a quadrilateral whose four sides all have the same length. In a rhombus, opposite sides are parallel, opposite angles are equal, and the diagonals are perpendicular bisectors of each other.

The word rhombus comes from the Greek rhombos, meaning a spinning top or bullroarer (a piece of wood whirled on a string). The plural is either rhombi or rhombuses — both are accepted.

A rhombus is also called a diamond in everyday language. The diamond suit on playing cards is rhombus-shaped. Less commonly, a rhombus is called a lozenge.

The Rhombus Family — How a Rhombus Relates to Other Shapes

A rhombus sits inside a hierarchy of quadrilaterals. Every rhombus is a parallelogram (its opposite sides are parallel). Every rhombus is also a kite (it has two pairs of equal adjacent sides — though in a rhombus, all four sides happen to be equal). A square is a rhombus where all four angles are 90°.

The table below compares a rhombus with the other common quadrilaterals:

The simplest way to place a rhombus: it is a parallelogram with all four sides equal, but with no requirement that the angles are right angles.

Properties of a Rhombus

A rhombus has seven core properties:

1. All four sides are equal in length.

2. Opposite sides are parallel.

3. Opposite angles are equal.

4. Adjacent angles are supplementary (they add up to 180°).

5. The diagonals bisect each other at right angles (90°).

6. The diagonals bisect the vertex angles.

7. The sum of the four interior angles is 360°.

A consequence of properties 5 and 6: the two diagonals divide the rhombus into four congruent right triangles.

Why Are the Diagonals Perpendicular?

The four triangles formed by the diagonals are congruent by SSS. Each triangle has two sides equal to half of each diagonal (because the diagonals bisect each other) and a side of the rhombus as its hypotenuse. Since all four sides of the rhombus are equal, the four triangles must be congruent — which forces the angle at the diagonal intersection to be 90°.

Diagonals of a Rhombus

A diagonal is a line segment joining two non-adjacent vertices. A rhombus has two diagonals. They have three defining behaviours:

• They are perpendicular to each other (they meet at 90°).

• They bisect each other (they cut each other exactly in half).

• They bisect the vertex angles.

The two diagonals are generally not equal in length. The diagonals of a rhombus are equal only when the rhombus is a square.

The Diagonal–Side Relationship

If a rhombus has side length a and diagonals of length d₁ and d₂:

d₁² + d₂² = 4a²

This follows from the Pythagorean theorem applied to one of the four right triangles formed by the diagonals. Each triangle has legs of length d₁/2 and d₂/2 and hypotenuse a. Squaring and combining gives the relationship above.

This formula is useful when a problem gives the diagonals and asks for the side, or gives the side and one diagonal and asks for the other.

Rhombus Formulas

A rhombus has two main formula categories: area (three methods) and perimeter (one formula).

Area of a Rhombus

Method 1 — Using the Diagonals

Area = (d₁ × d₂) / 2

Use this method when both diagonals are known. It is the most common formula in school problems.

Method 2 — Using the Base and Height

Area = b × h

Use this method when the side length and the height are both known.

Method 3 — Using the Side and an Angle

Area = a² × sin(θ)

Use this method when the side and one angle are known. This formula uses trigonometry and is typically introduced in Grade 9 onward.

Quick guide — which area formula to use:

Perimeter of a Rhombus

Perimeter = 4a

Since all four sides of a rhombus are equal, the perimeter is four times the side length.

How to Verify a Rhombus

A quadrilateral can be confirmed as a rhombus using any one of these five tests:

1. Side test: Show that all four sides are equal in length.

2. Diagonal test: Show that the diagonals bisect each other at right angles.

3. Parallelogram + adjacent-side test: Show it is a parallelogram with two adjacent sides equal.

4. Kite + parallel-sides test: Show it is a kite whose opposite sides are also parallel.

5. Coordinate-geometry test: For four given points, use the distance formula to confirm all four sides are equal, then use the slope formula or dot product to confirm the diagonals are perpendicular.

Any one of these is sufficient — they are equivalent characterisations of a rhombus.

Worked Examples

Example 1 — Area from Diagonals

Find the area of a rhombus with diagonals 10 cm and 8 cm.

Using Method 1:

Area = (d₁ × d₂) / 2 Area = (10 × 8) / 2 Area = 80 / 2 Area = 40 cm²

Example 2 — Perimeter from Side

Find the perimeter of a rhombus with side 7 cm.

Perimeter = 4a Perimeter = 4 × 7 Perimeter = 28 cm

Example 3 — Side from Diagonals

The diagonals of a rhombus are 16 cm and 12 cm. Find the side length.

Using d₁² + d₂² = 4a²:

16² + 12² = 4a² 256 + 144 = 4a² 400 = 4a² a² = 100 a = 10 cm

The side of the rhombus is 10 cm.

Rhombus in Real Life

The rhombus appears in many everyday and design contexts:

• The diamond suit on playing cards.

• Diamond-shaped traffic warning signs (used in the United States and several other countries to indicate road hazards ahead).

• The argyle pattern on socks, sweaters, and carpets.

• The rhombic lattice in crystallography — one of the five 2D lattice types.

• Some kite designs, when all four sides are made equal.

Not every diamond-shaped object is a true rhombus. A standard kite, for example, usually has two pairs of equal adjacent sides but unequal opposite sides — which makes it a kite, not a rhombus. For an object to be a true rhombus, all four sides must be equal in length.

Curriculum Reference

The rhombus appears in standard school curricula worldwide:

• NCERT (India) — Class 8 Chapter 3 Understanding Quadrilaterals; Class 9 Chapter 8 Quadrilaterals.

• Common Core State Standards (US) — 4.G.A.2 (classify two-dimensional figures based on side and angle properties); 5.G.B.3 (understand attributes of geometric shapes).

• UK National Curriculum — Key Stage 2 Year 5: Geometry — properties of shapes.

Common Mistakes

Three mistakes that come up routinely:

• Assuming the diagonals are equal. They aren't, in general. The diagonals of a rhombus are equal only when the rhombus is a square.

• Confusing a rhombus with a kite. A kite has two pairs of equal adjacent sides; a rhombus has all four sides equal. Every rhombus is a kite, but not every kite is a rhombus.

• Assuming all rhombuses have right angles. A rhombus has right angles only in the special case where it's a square. A general rhombus has two acute and two obtuse angles.

Related Terms