Area of a Triangle

Published on: April 12, 2026

This Area of a Triangle Calculator helps you find the area of the triangle formed by two 3D vectors. First calculate the cross product of the vectors, then find the magnitude of that result and divide by 2. This gives the area of the triangle. 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. Use Area = 1/2 · |A × B| and simplify.

Formula:

Area=
1
2
·|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 triangle area formula.

In this problem: Use Area = 1/2 · |A × B|.

Area=
1
2
·|A×B|

Step 3b - Compute the magnitude and multiply by 1/2.

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

Area=
1
2
·√( 0 )2+( 0 )2+( 1 )2

Step 3c - Simplify the result.

In this problem: Simplify the final expression if possible.

Area=
1
2

Final answer: Area = 1/2

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 triangle area formula.

In this problem: Use Area = 1/2 · |A × B|.

Area=
1
2
·|A×B|

Step 3b - Compute the magnitude and multiply by 1/2.

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

Area=
1
2
·√( -3 )2+( 6 )2+( -3 )2

Step 3c - Simplify the result.

In this problem: Simplify the final expression if possible.

Area=
3
2
√6

Final answer: Area = 3*sqrt(6)/2