Rhombus

What Is a Rhombus?

A rhombus is a quadrilateral with four sides of equal length. It is also a type of parallelogram, so each pair of opposite sides is parallel.

The following diagram represents a rhombus.

In the next diagram, the vertices are named A, B, C, and D. The four sides are AB, BC, CD, and DA, while AC and BD are the diagonals.

Rhombus Properties at a Glance

Element	Notation	Rhombus property
Vertices	A, B, C, D	A rhombus has four vertices.
Sides	AB, BC, CD, DA	AB = BC = CD = DA
Opposite sides	AB and CD; BC and DA	AB ∥ CD and BC ∥ DA
Angles	∠A, ∠B, ∠C, ∠D	∠A = ∠C, ∠B = ∠D, and adjacent angles add to 180°.
Diagonals	AC, BD	AC ⟂ BD, and the diagonals bisect each other.

Properties of a Rhombus Explained

All Four Sides of a Rhombus Are Equal

Equal side lengths are the defining property of a rhombus. For rhombus ABCD:

AB = BC = CD = DA

Opposite Angles of a Rhombus Are Equal

Because a rhombus is a parallelogram, each pair of opposite angles is equal:

∠A = ∠C and ∠B = ∠D.

Adjacent Angles in a Rhombus Add to 180 Degrees

Any two adjacent interior angles of a rhombus are supplementary. Therefore:

∠A + ∠B = 180°, ∠B + ∠C = 180°, ∠C + ∠D = 180°, and ∠D + ∠A = 180°.

In the diagram above, sides DA and BC are extended to make the angle relationships easier to see.

Opposite Sides of a Rhombus Are Parallel

A rhombus is a parallelogram with four equal sides. Its opposite sides are therefore parallel:

AB ∥ CD and BC ∥ DA.

The Two Rhombus Heights Have the Same Length

A rhombus can be viewed using either pair of opposite sides as its base. Since every side has the same length, the perpendicular distance between AB and CD equals the perpendicular distance between BC and DA. If these distances are p and q, then p = q.

Rhombus Diagonals Are Perpendicular Bisectors

The diagonals of a rhombus intersect at right angles and divide each other into equal halves. If diagonals AC and BD intersect at O, then:

• AC ⟂ BD
• AO = OC
• BO = OD

Each diagonal also bisects a pair of opposite angles. Diagonal AC bisects ∠A and ∠C, while diagonal BD bisects ∠B and ∠D.

Perimeter of a Rhombus

The perimeter of a rhombus is the total length of its four equal sides. If the side length is a, then:

Perimeter = 4a

Rhombus Perimeter Example

If each side of a rhombus is 7 cm, its perimeter is:

Perimeter = 4 × 7 = 28 cm

Area of a Rhombus

The area of a rhombus can be calculated from its base and height, from a side and an included angle, or from the lengths of its diagonals. Use the formula that matches the measurements given in the problem.

Rhombus Area Using Base and Perpendicular Height

If a is the side used as the base and h is the perpendicular height, then:

Area = a × h

Rhombus Area Using a Side and an Angle

If the side length is a and the included interior angle is θ, then:

Area = a² sin θ

For example, if a rhombus has side length 10 cm and an included angle of 30°, then its area is 10² × sin 30° = 100 × 0.5 = 50 cm².

Rhombus Area Using the Diagonals

If the diagonal lengths are d₁ and d₂, then:

Area = (d₁ × d₂) ÷ 2

For example, if the diagonals measure 12 cm and 8 cm, the area is (12 × 8) ÷ 2 = 48 cm².

Rhombus Compared with a Square and a Parallelogram

Shape	Side lengths	Angles	Diagonals
Rhombus	All four sides are equal.	Opposite angles are equal; angles are not necessarily 90°.	Perpendicular and bisect each other; not necessarily equal.
Square	All four sides are equal.	All four angles are 90°.	Perpendicular, equal, and bisect each other.
Parallelogram	Opposite sides are equal.	Opposite angles are equal.	Bisect each other; not generally perpendicular.

Every square is a rhombus because it has four equal sides. However, a rhombus is a square only when all four interior angles are 90°.

Common Questions About a Rhombus

Are the diagonals of a rhombus equal?

Not in general. The diagonals of a rhombus are perpendicular and bisect each other, but they are equal only in the special case where the rhombus is a square.

Does a rhombus always have four right angles?

No. A rhombus has two pairs of equal opposite angles. It has four right angles only when it is also a square.

How do you find a missing angle in a rhombus?

Use the facts that opposite angles are equal and adjacent angles add to 180°. For example, if one angle is 70°, the opposite angle is 70°, and each adjacent angle is 110°.

Can the area of a rhombus be found from one diagonal?

One diagonal alone is not enough. You need both diagonals, or another valid set of measurements such as a side and its perpendicular height.

Rhombus Formula Summary

• Perimeter: 4a
• Area from base and height: a × h
• Area from side and angle: a² sin θ
• Area from diagonals: (d₁ × d₂) ÷ 2

In this Geometry tutorial, we defined a rhombus, examined its sides, angles, and diagonals, and learned how to calculate its perimeter and area.