Rhombus Calculator

How this rhombus calculator works

A rhombus is a special parallelogram where all four sides are equal. Once you know any two independent measurements (for example side and height, or both diagonals), every other property of the rhombus is determined.

This calculator supports several common input combinations and automatically computes:

• side length \(a\)
• perimeter \(P\)
• area \(A\)
• height \(h\)
• diagonals \(d_1\) and \(d_2\)
• interior angles \(\alpha\) and \(\beta = 180^\circ - \alpha\)

Rhombus formulas

Basic properties

Diagonals from side and angle

In a rhombus, the diagonals are perpendicular and bisect each other. If \(a\) is the side and \(\alpha\) is an interior angle:

Side from diagonals

Each side is the hypotenuse of a right triangle with legs \(d_1/2\) and \(d_2/2\):

Height and angles

Supported input modes

1. Side and height

Given side \(a\) and height \(h\):

• Area: \(A = a \cdot h\)
• Perimeter: \(P = 4a\)
• \(\sin\alpha = h/a\), so \(\alpha = \arcsin(h/a)\)
• Diagonals from \(a\) and \(\alpha\) using the formulas above

2. Side and angle

Given side \(a\) and interior angle \(\alpha\):

• Height: \(h = a\sin\alpha\)
• Area: \(A = a^2\sin\alpha\)
• Perimeter: \(P = 4a\)
• Diagonals from \(a\) and \(\alpha\)

3. Both diagonals

Given diagonals \(d_1\) and \(d_2\):

• Area: \(A = \dfrac{d_1 d_2}{2}\)
• Side: \(a = \sqrt{(d_1/2)^2 + (d_2/2)^2}\)
• Perimeter: \(P = 4a\)
• \(\tan(\alpha/2) = d_2/d_1\), so \(\alpha = 2\arctan(d_2/d_1)\)
• Height: \(h = A/a\)

4. Side and one diagonal

If you know side \(a\) and diagonal \(d_1\):

• Half-diagonal: \(d_1/2\)
• Other half-diagonal: \(\dfrac{d_2}{2} = \sqrt{a^2 - (d_1/2)^2}\)
• So \(d_2 = 2\sqrt{a^2 - (d_1/2)^2}\)
• Then proceed as in the “both diagonals” case

5. Perimeter and area

Given perimeter \(P\) and area \(A\):

• Side: \(a = P/4\)
• Height: \(h = A/a\)
• \(\sin\alpha = h/a = 4A/P^2\)
• Then compute diagonals from \(a\) and \(\alpha\)

Worked example

Suppose a rhombus has side \(a = 6\ \text{cm}\) and height \(h = 4\ \text{cm}\).

1. Area: \[ A = a \cdot h = 6 \cdot 4 = 24\ \text{cm}^2 \]
2. Perimeter: \[ P = 4a = 4 \cdot 6 = 24\ \text{cm} \]
3. Angle: \[ \sin\alpha = \frac{h}{a} = \frac{4}{6} = \frac{2}{3} \Rightarrow \alpha \approx 41.81^\circ \]
4. Diagonals: \[ d_1 = a\sqrt{2 + 2\cos\alpha} \approx 6\sqrt{2 + 2\cdot 0.745} \approx 9.94\ \text{cm} \] \[ d_2 = a\sqrt{2 - 2\cos\alpha} \approx 6\sqrt{2 - 2\cdot 0.745} \approx 4.82\ \text{cm} \]

Common questions

Can a square be a rhombus?

Yes. A square is a special case of a rhombus where all angles are right angles (\(90^\circ\)). All the formulas above still apply.

What are typical units?

You can use any consistent length unit: millimeters, centimeters, meters, inches, feet, etc. The calculator treats inputs as abstract lengths and reports area in squared units of the same base unit (e.g., if you enter cm, area is in cm²).