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Parallelogram

A parallelogram has two pairs of parallel sides. Area = base × height, or a×b×sin(θ), or ½d₁d₂sin(φ). Diagonals bisect each other; parallelogram law: d₁²+d₂²=2(a²+b²).

Concept Fundamentals
A = b × h
Area (base×h)
A = a×b×sin(θ)
Area (sides)
P = 2(a + b)
Perimeter
d₁²+d₂²=2(a²+b²)
Law

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A rectangle is a parallelogram with all angles 90°. A rhombus has all sides equal. The parallelogram law: sum of two vectors forms the diagonal of the parallelogram they span. A square is a parallelogram (and rectangle and rhombus) with equal sides and 90° angles.

Key quantities
A = b × h
Area (base×h)
Key relation
A = a×b×sin(θ)
Area (sides)
Key relation
P = 2(a + b)
Perimeter
Key relation
d₁²+d₂²=2(a²+b²)
Law
Key relation

Ready to run the numbers?

Why: Parallelograms appear in floor tiles, solar panels, and truss bridges. The parallelogram law of vector addition uses the diagonal for sum.

How: Area = base × height (perpendicular). With sides and angle: A = a×b×sin(θ). With diagonals: A = ½d₁d₂sin(φ). Height is always perpendicular to base.

A rectangle is a parallelogram with all angles 90°. A rhombus has all sides equal.The parallelogram law: sum of two vectors forms the diagonal of the parallelogram they span.
Sources:Wikipedia - ParallelogramWolfram MathWorld - Parallelogram

Run the calculator when you are ready.

Parallelogram CalculatorEnter dimensions in base+height, sides+angle, or diagonals+angle mode
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PARALLELOGRAM

▱ Parallelogram Calculator — Area, Perimeter & Diagonals

Enter dimensions in any of three modes: Base & Height, Sides & Angle, or Diagonals. Results update automatically with step-by-step breakdown.

Enter Dimensions

Please enter valid positive numbers for base and height.

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

🧮 Fascinating Math Facts

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Parallelogram area = base × height — always use perpendicular height.

— Formula

90°

At 90° included angle, the parallelogram becomes a rectangle.

— Special case

📋 Key Takeaways

  • • Parallelogram: quadrilateral with two pairs of parallel sides
  • • Area = base × height, or atimesbtimessin(theta)a \\times b \\times \\sin(\\theta), or frac12d1d2sin(phi)\\frac{1}{2} d_1 d_2 \\sin(\\phi)
  • • Perimeter = 2(a + b)
  • • Diagonals bisect each other; parallelogram law: d12+d22=2(a2+b2)d_1^2 + d_2^2 = 2(a^2 + b^2)

💡 Did You Know?

📐A rectangle is a parallelogram with all angles 90°. A rhombus is a parallelogram with all sides equal. A square is both.
🏗️Parallelogram shapes appear in floor tiles, solar panel arrays, and truss bridges—the parallel sides distribute load efficiently.
📏The parallelogram law of vector addition: the sum of two vectors forms the diagonal of the parallelogram they span.
🔷Euclid defined parallelograms in Book I of the Elements. Opposite sides and angles are equal; consecutive angles sum to 180°.
◇The area formula A = base × height works because any parallelogram can be cut and rearranged into a rectangle of the same area.
📝In coordinate geometry, the area of a parallelogram with vertices can be computed using the cross product of two adjacent side vectors.

📖 How Parallelogram Calculations Work

A parallelogram has two pairs of parallel sides. Choose the input mode that matches your given dimensions.

Mode 1: Base & Height

Enter base (b) and perpendicular height (h). Area = b × h. Assumes a rectangle for perimeter (P = 2(b + h)).

Mode 2: Sides & Angle

Enter sides a, b and included angle θ. Area = a × b × sin(θ). Height = b × sin(θ). Diagonals from law of cosines.

Mode 3: Diagonals & Angle

Enter diagonals d₁, d₂ and angle φ between them. Area = ½ d₁ d₂ sin(φ). Sides derived from parallelogram law.

🎯 Expert Tips

Height vs Slant

Height is always perpendicular to the base. Never use the slanted side length as height.

Angle Range

Included angle must be 0° < θ < 180°. At 90° you have a rectangle.

Parallelogram Law

d₁² + d₂² = 2(a² + b²) relates diagonals to sides. Useful for vector addition and mechanics.

Unit Consistency

Use the same units for all inputs. Area will be in square units; perimeter in linear units.

📊 Parallelogram vs Rectangle vs Rhombus vs Trapezoid

ShapeParallel SidesSpecial Properties
Parallelogram2 pairsOpposite sides equal, diagonals bisect
Rectangle2 pairsAll angles 90°, diagonals equal
Rhombus2 pairsAll sides equal, diagonals perpendicular
Trapezoid1 pairOnly one pair of parallel sides

❓ Frequently Asked Questions

What is a rectangle?

A rectangle is a parallelogram with all angles 90°. Area = length × width; perimeter = 2(length + width).

What is a rhombus?

A rhombus is a parallelogram with all four sides equal. Diagonals are perpendicular and bisect each other.

Can I use the slanted side as height?

No. Height must be the perpendicular distance from the base to the opposite side. Using the slanted side gives wrong area.

What angle range is valid?

The included angle θ must satisfy 0° < θ < 180°. At 90° the parallelogram is a rectangle.

How do diagonals relate to sides?

Parallelogram law: d₁² + d₂² = 2(a² + b²). The diagonals bisect each other at their midpoint.

Is a square a parallelogram?

Yes. A square is a parallelogram (and a rectangle and rhombus) with all sides equal and all angles 90°.

What units should I use?

Use any consistent linear units (mm, cm, m, in, ft, yd). Area will be in square units; perimeter in the same linear units.

📈 Parallelogram by the Numbers

3
Input Modes
2
Pairs of Parallel Sides
180°
Consecutive Angles Sum
∞
Real-World Uses

Disclaimer: This calculator provides mathematically precise results based on standard geometric formulas. Results are limited by floating-point precision (~15 significant digits). For critical applications, verify with domain-specific tools. Not a substitute for professional analysis.

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