Rhombus: Properties, Area, and Perimeter

Geometry
BT
Last updated on June 9, 2026

Medieval architects tiled the Alhambra Palace using thousands of interlocking rhombuses.

Medieval Islamic architects had no computers, no CAD software, and no industrial tile cutters. Working with compasses and straightedges, they generated complex geometric tilings across walls, floors, and ceilings — and at the heart of almost every pattern is the rhombus. The shape tiles the plane perfectly because of two properties it never gives up: all four sides equal, and opposite sides parallel. Change either, and the tiling falls apart.

A rhombus is a quadrilateral in which all four sides are equal in length. This single rule gives it a surprisingly rich set of properties — and two different ways to calculate area, depending on what measurements are known.

Properties of A Rhombus

All of these properties follow from the single defining rule: four equal sides.

• All four sides are equal: AB=BC=CD=DA=a
• Opposite sides are parallel: The rhombus is a parallelogram — opposite sides are parallel and opposite angles are equal.
• Opposite angles are equal: ∠A=∠C and ∠B=∠D
• Adjacent angles are supplementary: Any two angles at adjacent vertices add up to 180°: ∠A+∠B=180°
• Diagonals bisect each other at 90°: The two diagonals cut each other into two equal halves, and the angle between them is always exactly 90°.
• Diagonals bisect the vertex angles: Each diagonal divides its two vertex angles into two equal parts.
• Diagonals create four congruent right triangles: The intersection of the diagonals produces four right triangles, each with hypotenuse a and legs ( \frac{d_1}{2} ) and ( \frac{d_2}{2} ).

The relationship between the side and diagonals follows from the Pythagorean theorem:

a²=(d₁/2)²+(d₂/2)²

Rhombus vs. Square vs. Parallelogram

These three shapes are often confused. Here is the hierarchy:

• Every square is a rhombus (four equal sides) AND a rectangle (four right angles).
• Every rhombus is a parallelogram (opposite sides parallel), but not every rhombus is a square.
• A rhombus with all right angles = square. A rhombus without right angles = rhombus only.

The diagonals of a square are equal in length. The diagonals of a general rhombus are not equal. That asymmetry — the unequal diagonals — is what creates the rhombus's characteristic "diamond lean."

The Two Area Formulas And Why They Both Work

Method 1: Base × Height

A=b×h

where b is any side of the rhombus (all four sides are equal, so any side works as the base) and h is the perpendicular distance between two opposite parallel sides.

This works because a rhombus is a parallelogram, and the area of a parallelogram is base × height.

Method 2: Diagonal formula (more commonly used)

A=( \frac{d_1 \times d_2}{2} )

where d₁ and d₂ are the lengths of the two diagonals.

Why does this formula work? Here is the visual proof.

The two diagonals divide the rhombus into four congruent right triangles. Each triangle has legs of ( \frac{d_1}{2} ) and ( \frac{d_2}{2} ), so the area of each triangle is:

Area of one triangle=( \frac{1}{2} \times \frac{d_1}{2} \times \frac{d_2}{2} = \frac{d_1 d_2}{8} )

There are four such triangles, so the total area is:

A=4×( \frac{d_1 d_2}{8} = \frac{d_1 d_2}{2} )

Alternatively: draw a rectangle that exactly encloses the rhombus, with sides d₁ and d₂. The rhombus fills exactly half of that rectangle. So its area is ( \frac{d_1 \times d_2}{2} ).

Perimeter of a Rhombus

Since all four sides are equal:

P=4a

where a is the length of one side.

If the diagonals are known instead of the side length, use the Pythagorean theorem to find a first:

a=√( ( ( \frac{d_1}{2} ))²+( ( \frac{d_2}{2} ))² )

Then P=4a.

Worked Examples of Rhombus

Example 1: Area from diagonals

A rhombus has diagonals of length 10 cm and 24 cm. Find the area and perimeter.

Area:

A=( \frac{10 \times 24}{2} = 120 \text{ cm}^2 )

Perimeter (find side first):

a=√( ( ( \frac{10}{2} ))²+( ( \frac{24}{2} ))² ) = 13

P=4×13=52 cm

Final answers: Area = 120 cm², Perimeter = 52 cm

Example 2: Area from base and height

A rhombus has a side of 9 m and a perpendicular height of 7 m. Find the area.

A=b×h=9×7=63 m²

Final answer: 63 m²

Example 3: Finding a missing diagonal

A rhombus has area 84 cm² and one diagonal of length 12 cm. Find the other diagonal.

Using A=( \frac{d_1 \times d_2}{2} ) and rearranging:

84=( \frac{12 \times d_2}{2} ) = 6d₂

d₂=14 cm

Final answer: The other diagonal is 14 cm.

The Mathematicians Behind The Rhombus

The rhombus has been a practical shape throughout mathematics history. Euclid of Alexandria (c. 300 BCE) defined and discussed it in Elements, classifying it among the quadrilaterals. But the shape's deepest mathematical treatment came from the medieval Islamic world.

Abū al-Wafāʾ al-Būzjānī (940–998 CE, Persia) was a Persian mathematician who wrote A Book on Those Geometric Constructions Which Are Necessary for a Craftsman — a practical guide to geometric constructions for architects and artisans. His work on dividing and arranging rhombuses, squares, and triangles formed the mathematical foundation for the complex geometric tilework seen in mosques, madrasas, and palaces across the Islamic world, including the Alhambra.

Common Mistakes With Rhombuses

Mistake 1: Using d₁ × d₂ without the ½ (forgetting to halve)

A quick mental check: the rhombus's area should be smaller than the rectangle that encloses it. If your answer equals the rectangle's full area, you have skipped the ½.

Mistake 2: Using the side length as the height

The height depends on the angle: h=asin⁡θ where θ is one of the vertex angles.

Mistake 3: Assuming both diagonals are equal

Equal diagonals only occur in a square. Treat d₁ and d₂ as separate values unless the problem explicitly states the shape is a square.

Quick Reference

Next Steps

Find the area and perimeter of a rhombus with diagonals 16 cm and 12 cm. Then try: a rhombus has side 10 cm and one diagonal of 16 cm — find the other diagonal and the area.