Rhombus Calculator
Calculate area, perimeter, diagonals, and angles of any rhombus
All sides of a rhombus are equal
The other diagonal will be calculated
Diagonals of a rhombus bisect each other at 90°
Any interior angle of the rhombus (opposite angles are equal)
Rhombus Properties
Area
Perimeter
Side Length
Height
Diagonal d₁ (longer)
Diagonal d₂ (shorter)
Acute Angle
°
Opposite angles are equal
Obtuse Angle
°
Adjacent angles sum to 180°
Rhombus Properties
Rhombus Formulas
Area (from diagonals)
A = (d₁ × d₂) / 2
Perimeter
P = 4a
Area (from angle)
A = a² × sin(θ)
Diagonal Relationship
d₁² + d₂² = 4a²
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About Rhombus Calculator
What is a Rhombus?
A rhombus is a special type of parallelogram where all four sides are equal in length. It is also known as a diamond or lozenge shape. The opposite angles of a rhombus are equal, and adjacent angles are supplementary (add up to 180°).
Rhombuses are commonly found in playing card suits (diamonds), kites, jewelry designs, and various architectural patterns.
Key Properties of a Rhombus
All Sides are Equal
The defining characteristic of a rhombus is that all four sides have the same length. This distinguishes it from a general parallelogram.
Diagonals Bisect Each Other at Right Angles
Unlike a general parallelogram, the diagonals of a rhombus intersect at 90° angles. They also bisect each other (cut each other in half).
Diagonals Bisect the Angles
Each diagonal divides the corner angles into two equal parts.
Opposite Angles are Equal
Like all parallelograms, opposite angles in a rhombus are equal.
How to Use This Calculator
From Side & Diagonal Mode
- Select Side & Diagonal mode
- Enter the side length (a)
- Enter one diagonal length (d₁)
- The calculator computes the other diagonal, area, perimeter, height, and angles
From Diagonals Mode
- Select Diagonals mode
- Enter both diagonal lengths (d₁ and d₂)
- The calculator derives side length, area, perimeter, and angles
From Side & Angle Mode
- Select Side & Angle mode
- Enter the side length (a)
- Enter an interior angle (in degrees)
- Get complete rhombus measurements including diagonals and height
Rhombus Formulas
Core Formulas
| Property | Formula | Description |
|---|---|---|
| Area (from diagonals) | A = (d₁ × d₂) / 2 | Half the product of diagonals |
| Area (from side × height) | A = a × h | Side times perpendicular height |
| Area (from side × sin) | A = a² × sin(θ) | Side squared times sine of angle |
| Perimeter | P = 4a | Four times the side length |
| Height | h = A / a | Area divided by side |
Diagonal Formulas
| Property | Formula | |----------|---------|| | Diagonal from side & angle | d₁ = a × √(2 + 2cos(θ)) | | Diagonal from side & angle | d₂ = a × √(2 - 2cos(θ)) | | Side from diagonals | a = √((d₁/2)² + (d₂/2)²) | | Diagonal relationship | d₁² + d₂² = 4a² |
Angle Formulas
| Property | Formula | |----------|---------|| | Adjacent angles | θ₁ + θ₂ = 180° | | Angle from diagonals | θ = 2 × arctan(d₂/d₁) |
Rhombus vs Other Quadrilaterals
Rhombus vs Parallelogram
A rhombus is a special parallelogram where all sides are equal. In a general parallelogram, only opposite sides are equal.
Rhombus vs Square
A square is a special rhombus where all angles are 90°. A rhombus can have any angle configuration as long as opposite angles are equal.
Rhombus vs Kite
Both have perpendicular diagonals, but a kite has two pairs of adjacent equal sides, while a rhombus has all four sides equal.
Real-World Applications
Design & Art
- Diamond patterns in textiles and wallpaper
- Playing card suit symbols
- Jewelry and gemstone cuts
- Tile and mosaic patterns
Architecture
- Window designs
- Decorative elements
- Floor patterns
Engineering
- Mechanical linkages
- Structural analysis
- Folding mechanisms
Frequently Asked Questions
Is a square a rhombus?
Yes! A square is a special type of rhombus where all angles are exactly 90°. Every square is a rhombus, but not every rhombus is a square.
Why do the diagonals of a rhombus intersect at 90°?
This is a unique property of rhombuses (and squares). Because all sides are equal, the triangles formed by the diagonals are congruent, which forces the diagonals to meet at right angles.
How is a rhombus different from a diamond?
"Diamond" is often used as an informal name for a rhombus, especially when referring to the shape oriented so that one diagonal is vertical (like a playing card diamond suit).
Can a rhombus have a 90° angle?
Yes, if a rhombus has one 90° angle, then all angles must be 90° (making it a square). This is because opposite angles are equal and adjacent angles are supplementary.
What is the relationship between the diagonals?
In a rhombus, the diagonals are perpendicular bisectors of each other. Also, d₁² + d₂² = 4a² (where a is the side length).
Important Notes
- All sides of a rhombus must be equal in length
- The diagonals always intersect at right angles (90°)
- Interior angles must be between 0° and 180° (exclusive)
- The sum of all interior angles is always 360°
- If all angles are 90°, the rhombus becomes a square
Note: This calculator provides mathematical results for geometric shapes. For engineering or construction projects, always consult with qualified professionals and verify measurements.