Area of a Parallelogram Calculator

Published on: April 5, 2026

This Area of a Parallelogram Calculator helps you find the area of the parallelogram formed by two 3D vectors. First calculate the cross product of the vectors, then find the magnitude of that result. The magnitude gives the area of the parallelogram. It is a simple way to check answers, understand the method clearly, and practise vector operations step by step.

Step-by-step method

  1. Identify the vector components.
  2. Compute the cross product A × B.
  3. Compute the magnitude |A × B| to get the area.

Formula:

Area=|A×B|
A×B=( a2·b3−a3·b2, a3·b1−a1·b3, a1·b2−a2·b1 )

Example 1: (1,0,0),(0,1,0)

Step 1 - Identify the components.

In this problem: From the given vectors A = (1, 0, 0) and B = (0, 1, 0), the components are:

a1=1, a2=0, a3=0
b1=0, b2=1, b3=0

Step 2a - Write the cross product formula.

In this problem: Use the component formula for A × B.

A×B=( a2·b3−a3·b2, a3·b1−a1·b3, a1·b2−a2·b1 )

Step 2b - Substitute values.

In this problem: Substitute the components into the cross product formula.

A×B=( ( 0 )·( 0 )−( 0 )·( 1 ), ( 0 )·( 0 )−( 1 )·( 0 ), ( 1 )·( 1 )−( 0 )·( 0 ) )

Step 2c - Multiply and subtract.

In this problem: Compute the cross product components.

A×B=( 0−0, 0−0, 1−0 )=( 0, 0, 1 )

Step 3a - Write the area formula.

In this problem: Area = |A × B|.

Area=|A×B|

Step 3b - Use the computed cross product.

In this problem: Replace A × B with the vector from Step 2c.

Area=|A×B|

Step 3c - Compute the magnitude.

In this problem: Use |(x,y,z)| = √(x² + y² + z²).

Area=√( 0 )2+( 0 )2+( 1 )2

Step 3d - Simplify the result.

In this problem: No further simplification is needed.

Area=1

Final answer: Area = 1

Example 2: (1,2,3),(4,5,6)

Step 1 - Identify the components.

In this problem: From the given vectors A = (1, 2, 3) and B = (4, 5, 6), the components are:

a1=1, a2=2, a3=3
b1=4, b2=5, b3=6

Step 2a - Write the cross product formula.

In this problem: Use the component formula for A × B.

A×B=( a2·b3−a3·b2, a3·b1−a1·b3, a1·b2−a2·b1 )

Step 2b - Substitute values.

In this problem: Substitute the components into the cross product formula.

A×B=( ( 2 )·( 6 )−( 3 )·( 5 ), ( 3 )·( 4 )−( 1 )·( 6 ), ( 1 )·( 5 )−( 2 )·( 4 ) )

Step 2c - Multiply and subtract.

In this problem: Compute the cross product components.

A×B=( 12−15, 12−6, 5−8 )=( -3, 6, -3 )

Step 3a - Write the area formula.

In this problem: Area = |A × B|.

Area=|A×B|

Step 3b - Use the computed cross product.

In this problem: Replace A × B with the vector from Step 2c.

Area=|A×B|

Step 3c - Compute the magnitude.

In this problem: Use |(x,y,z)| = √(x² + y² + z²).

Area=√( -3 )2+( 6 )2+( -3 )2

Step 3d - Simplify the result.

In this problem: No further simplification is needed.

Area=3√6

Final answer: Area = 3*sqrt(6)