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Rhombus

Rhombus

A parallelogram with four congruent sides. Note that the diagonals of a rhombus are perpendicular (as is the case with all kites).

Note: A square is a special kind of rhombus.

 

Rhombus

s = side length of rhombus
h = height of rhombus
d1 = long diagonal of rhombus
d2 = short diagonal of rhombus

Area = hs
         = s2 sin A
         = s2 sin B
         = (½) d1d2


Rhombus with vertices A, B, C, D; side length s labeled on all sides; height h shown as dotted vertical line with right angle...

Rhombus with vertices A, B, C, D and perpendicular diagonals labeled d1 (long) and d2 (short) intersecting at center point O.

 

See also

Quadrilateral, trapezoid, area of a rhombus, sine

Key Formula

A=h⋅s=s2sin⁡A=s2sin⁡B=12 d1 d2A = h \cdot s = s^2 \sin A = s^2 \sin B = \tfrac{1}{2}\,d_1\,d_2
Where:
  • AA = Area of the rhombus (also used as an interior angle label in the sine formulas)
  • ss = Side length of the rhombus (all four sides are equal)
  • hh = Height (altitude) of the rhombus — the perpendicular distance between two parallel sides
  • d1d_1 = Length of the longer diagonal
  • d2d_2 = Length of the shorter diagonal
  • sin⁡A,sin⁡B\sin A, \sin B = Sine of either interior angle; since consecutive angles are supplementary, sin A = sin B

Worked Example

Problem: A rhombus has diagonals of length 10 cm and 24 cm. Find its area and side length.
Step 1: Use the diagonal formula to find the area.
A=12 d1 d2=12(24)(10)=120 cm2A = \tfrac{1}{2}\,d_1\,d_2 = \tfrac{1}{2}(24)(10) = 120 \text{ cm}^2
Step 2: Recall that the diagonals of a rhombus bisect each other at right angles. This creates four right triangles, each with legs half the length of each diagonal.
Half-diagonals: 242=12 cm,102=5 cm\text{Half-diagonals: } \tfrac{24}{2} = 12 \text{ cm}, \quad \tfrac{10}{2} = 5 \text{ cm}
Step 3: Apply the Pythagorean theorem to one of these right triangles to find the side length.
s=122+52=144+25=169=13 cms = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \text{ cm}
Answer: The area is 120 cm² and the side length is 13 cm.

Another Example

This example uses the angle-based area formula instead of the diagonal formula, and shows how to find diagonal lengths from the side and angle — a common exam variation.

Problem: A rhombus has a side length of 10 cm and one interior angle of 60°. Find its area and the length of each diagonal.
Step 1: Use the sine formula to find the area.
A=s2sin⁡(60°)=102⋅32=100⋅32=503≈86.6 cm2A = s^2 \sin(60°) = 10^2 \cdot \frac{\sqrt{3}}{2} = 100 \cdot \frac{\sqrt{3}}{2} = 50\sqrt{3} \approx 86.6 \text{ cm}^2
Step 2: The diagonals bisect the interior angles. The 60° angle is split into two 30° angles, and the opposite 120° angle is split into two 60° angles. In the right triangle formed at the center, the legs are s sin 30° and s cos 30°, which give half of each diagonal.
d12=ssin⁡(60°)=10⋅32=53,d22=ssin⁡(30°)=10⋅12=5\tfrac{d_1}{2} = s \sin(60°) = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}, \qquad \tfrac{d_2}{2} = s \sin(30°) = 10 \cdot \frac{1}{2} = 5
Step 3: Double each half-diagonal to get the full diagonal lengths.
d1=103≈17.32 cm,d2=10 cmd_1 = 10\sqrt{3} \approx 17.32 \text{ cm}, \qquad d_2 = 10 \text{ cm}
Step 4: Verify with the diagonal area formula.
12 d1 d2=12(103)(10)=503≈86.6 cm2  ✓\tfrac{1}{2}\,d_1\,d_2 = \tfrac{1}{2}(10\sqrt{3})(10) = 50\sqrt{3} \approx 86.6 \text{ cm}^2 \; \checkmark
Answer: The area is 503≈86.650\sqrt{3} \approx 86.6 cm². The diagonals are 103≈17.3210\sqrt{3} \approx 17.32 cm and 10 cm.

Frequently Asked Questions

What is the difference between a rhombus and a square?
A square is a special type of rhombus where all four interior angles are 90°. Every square is a rhombus, but a rhombus is only a square when its angles are all right angles. In a general rhombus, opposite angles are equal but not necessarily 90°.
Are the diagonals of a rhombus equal?
No. The diagonals of a rhombus are perpendicular bisectors of each other, but they are generally different lengths. They are equal only when the rhombus is also a square. The longer diagonal connects the two obtuse-angle vertices, and the shorter diagonal connects the two acute-angle vertices.
Is every rhombus a parallelogram?
Yes. A rhombus has two pairs of parallel sides, so it satisfies the definition of a parallelogram. It inherits all parallelogram properties: opposite sides are parallel and equal, opposite angles are equal, and the diagonals bisect each other. The rhombus adds the extra condition that all four sides are equal.

Rhombus vs. Rectangle

RhombusRectangle
SidesAll four sides are congruentOpposite sides are congruent (adjacent sides may differ)
AnglesOpposite angles are equal; consecutive angles are supplementary (not necessarily 90°)All four angles are 90°
DiagonalsPerpendicular but generally unequal in lengthEqual in length but generally not perpendicular
Area formulas2sin⁡As^2 \sin A or 12d1d2\tfrac{1}{2}d_1 d_2length × width
Special overlapA rhombus that is also a rectangle is a squareA rectangle that is also a rhombus is a square

Why It Matters

Rhombuses appear frequently in geometry courses when studying quadrilateral classification, coordinate proofs, and area problems. Many standardized tests ask you to distinguish a rhombus from other parallelograms or to compute its area using diagonals. The perpendicular-diagonal property is also essential in constructions and in understanding kite geometry, since every rhombus is a kite.

Common Mistakes

Mistake: Assuming the diagonals of a rhombus are equal in length.
Correction: The diagonals are perpendicular bisectors of each other, but they have different lengths unless the rhombus is a square. Always check whether the problem states equal diagonals before treating them as such.
Mistake: Using base × height with the slant side as the height.
Correction: The height h is the perpendicular distance between two parallel sides, not the side length itself. Area = s × h only when h is the altitude. If you know an angle instead, use Area = s² sin A.

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