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Rhombus

A rhombus is a quadrilateral with four equal sides—a special parallelogram. Area = ½d₁d₂, or s²sin(θ), or base × height. Diagonals bisect at 90°.

Concept Fundamentals
A = ½d₁d₂
Area (diagonals)
A = s²sin(θ)
Area (side & angle)
P = 4s
Perimeter
θ₁ + θ₂ = 180°
Angles

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A square is a rhombus with all angles 90°. Every square is a rhombus. Baseball infields are diamond-shaped—the diagonal formula applies. Diagonals of a rhombus are perpendicular and bisect each other.

Key quantities
A = ½d₁d₂
Area (diagonals)
Key relation
A = s²sin(θ)
Area (side & angle)
Key relation
P = 4s
Perimeter
Key relation
θ₁ + θ₂ = 180°
Angles
Key relation

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Why: Rhombuses appear in diamond rings, baseball diamonds, kites, and parquet flooring. The diagonals formula is simplest when both are known.

How: Use diagonals: A = ½d₁d₂. Use side and angle: A = s²sin(θ). Use side and height: A = s×h. Perimeter is always 4s.

A square is a rhombus with all angles 90°. Every square is a rhombus.Baseball infields are diamond-shaped—the diagonal formula applies.
Sources:Wikipedia - RhombusMath Is Fun - Rhombus

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Rhombus CalculatorEnter dimensions in any mode: diagonals, side+angle, side+height, or side+diagonal
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Rhombus Calculator

Calculate area, perimeter, diagonals, angles. Use diagonals, side+angle, side+height, or side+diagonal. Results update automatically.

Calculator Inputs

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

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Rhombus area = ½d₁d₂ — half the product of diagonals.

— Formula

90°

Diagonals of a rhombus intersect at right angles.

— Property

📋 Key Takeaways

  • • A rhombus has four equal sides; it is a special parallelogram
  • • Area: A=fracd1d22A = \\frac{d_1 d_2}{2} or A=s2sin(theta)A = s^2 \\sin(\\theta) or A=stimeshA = s \\times h
  • • Perimeter: P=4sP = 4s; opposite angles equal, consecutive supplementary (180°)
  • • Diagonals bisect each other at 90° and bisect vertex angles

💡 Did You Know?

💎A square is a special rhombus with all angles 90°. Every square is a rhombus, but not every rhombus is a square.Source: Geometry
⚾A baseball diamond is a rhombus (actually a square). The infield area uses the diagonal formula.Source: Sports
🪁Traditional kites often have rhombus shapes. The diagonals formula applies when you know both diagonal lengths.Source: Recreation
📐The diagonals of a rhombus are perpendicular. Their lengths determine the area: A = ½d₁d₂.Source: Euclidean geometry
🔷Rhombus tiles are used in parquet flooring and decorative patterns. Area helps estimate material needed.Source: Design
📏With side and one angle, use A = s² sin(θ). With side and height, use A = s × h (base × height).Source: Trigonometry

📖 How Rhombus Calculations Work

A rhombus is a quadrilateral with all four sides equal. Opposite angles are equal; consecutive angles sum to 180°. The diagonals bisect each other at right angles.

Area Formulas

A=fracd1timesd22A = \\frac{d_1 \\times d_2}{2} (diagonals). A=s2sin(theta)A = s^2 \\sin(\\theta) (side & angle). A=stimeshA = s \\times h (base × height).

Perimeter & Angles

P=4sP = 4s. theta1+theta2=180°\\theta_1 + \\theta_2 = 180°. From diagonals: s=sqrt(d1/2)2+(d2/2)2s = \\sqrt{(d_1/2)^2 + (d_2/2)^2}.

🎯 Expert Tips

Use Diagonals When Known

A = ½d₁d₂ is the simplest. Measure both diagonals and multiply, then halve.

Side & Angle for Jewelry

Diamond cuts often give side and angle. Use A = s² sin(θ). Ensure angle is in degrees.

Height ≤ Side

In side-height mode, height must be ≤ side (perpendicular distance). If height > side, check units.

Diagonal < 2× Side

In side-diagonal mode, diagonal must be less than twice the side. Otherwise the shape is invalid.

⚖️ Rhombus vs Other Shapes

ShapeEqual SidesRight AnglesArea Formula
Rhombus4No (unless square)½d₁d₂ or s²sin(θ)
Square4Yess²
Rectangle2 pairsYesl × w
Parallelogram2 pairsNob × h

❓ Frequently Asked Questions

What is the difference between a rhombus and a square?

A square is a rhombus with all angles 90°. Every square is a rhombus; a rhombus is a square only when all angles are right angles.

When should I use diagonals vs side and angle?

Use diagonals when you can measure both diagonals (e.g., from a diagram). Use side and angle when you have one side and one interior angle (e.g., diamond cut specifications).

Can a rhombus have a 90° angle?

Yes. If all four angles are 90°, it is a square. If only one pair of opposite angles is 90°, it is still a rhombus (but not a square).

How do I find the second diagonal from side and one diagonal?

Use d₂ = √(4s² − d₁²). The calculator does this automatically in side-diagonal mode.

What units should I use?

Use consistent units (e.g., all cm or all m). Area will be in squared units (cm², m²).

Why does height have to be ≤ side?

Height is the perpendicular distance from a side to the opposite side. It cannot exceed the side length in a valid rhombus.

How do diagonals relate to the area?

Area = ½ × d₁ × d₂. The diagonals are perpendicular and bisect each other. Their product divided by 2 gives the area.

📊 Rhombus by the Numbers

4
Equal sides
180°
Consecutive angles sum
90°
Diagonals intersect
P=4s
Perimeter formula

⚠️ Disclaimer: This calculator is for educational use. Verify critical measurements with professional tools. Results assume a flat, Euclidean rhombus.

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