Mathwords logoMathwords

diagonals of a rhombus — Definition, Formula & Examples

The diagonals of a rhombus are the two line segments connecting opposite vertices. They always cross at right angles and cut each other exactly in half.

In a rhombus with vertices AA, BB, CC, DD, the diagonals AC‾\overline{AC} and BD‾\overline{BD} are perpendicular bisectors of each other. If their lengths are d1d_1 and d2d_2, the intersection point divides each diagonal into two equal segments of length d12\tfrac{d_1}{2} and d22\tfrac{d_2}{2}, respectively.

Key Formula

A=d1⋅d22A = \frac{d_1 \cdot d_2}{2}
Where:
  • AA = Area of the rhombus
  • d1d_1 = Length of one diagonal
  • d2d_2 = Length of the other diagonal

How It Works

Because all four sides of a rhombus are equal, each diagonal splits the rhombus into two congruent isosceles triangles. The two diagonals together split it into four congruent right triangles. At the center, the diagonals meet at 90°90°, and each half-diagonal and a side form a right triangle you can solve with the Pythagorean theorem. This perpendicular-bisector property is also the reason the area of a rhombus equals half the product of its diagonals.

Worked Example

Problem: A rhombus has diagonals of length 10 cm and 24 cm. Find the side length and the area.
Find the half-diagonals: The diagonals bisect each other, so the half-lengths are 5 cm and 12 cm.
d12=5,d22=12\frac{d_1}{2} = 5, \quad \frac{d_2}{2} = 12
Use the Pythagorean theorem for the side: Each side is the hypotenuse of a right triangle formed by the two half-diagonals.
s=52+122=25+144=169=13 cms = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \text{ cm}
Calculate the area: Apply the diagonal-based area formula.
A=10×242=120 cm2A = \frac{10 \times 24}{2} = 120 \text{ cm}^2
Answer: The side length is 13 cm and the area is 120 cm².

Why It Matters

The perpendicular-bisector property shows up in coordinate geometry proofs and in engineering contexts where diamond-shaped cross-sections appear. In courses like geometry and precalculus, you will use it to derive side lengths, angles, and areas without needing trigonometry.

Common Mistakes

Mistake: Assuming the diagonals of a rhombus are equal in length.
Correction: The diagonals are equal only if the rhombus is also a square. In a general rhombus the two diagonals have different lengths.

Related Terms