Using Determinant to find the Area of a Parallelogram

Find Area Using Determinant
Determinants can be used to calculate the area of a parallelogram formed by 2 two-dimensional vectors. The matrix made from these two vectors has a determinant equal to the area of the parallelogram.
Area determinants are quick and easy to solve if you know how to solve a 2×2 determinant. A parallelogram in three dimensions is found using the cross product.

The following diagram shows how to find the area of a parallelogram formed by two vectors using determinant. Scroll down the page for more examples and solutions.

 

How to find the area of a parallelogram using determinant?

1. Represent the Sides as Vectors:
   Let the two adjacent sides of the parallelogram be represented by vectors u = (u₁, u₂) and v = (v₁, v₂).
   

2. Form a 2x2 Matrix:
   Create a 2x2 matrix where the components of the vectors u and v form the rows (or columns) of the matrix.
   If the parallelogram is represented by vectors u = (u₁, u₂) and v = (v₁, v₂) then form the matrix:
   
   
   
   

3. Calculate the Determinant:
   The determinant of matrix M is: det(M) = (u₁v₂) - (u₂v₁)
   

4. Find the Absolute Value:
   The area of the parallelogram is the absolute value of the determinant calculated in the previous step.
   Area = |det(M)| = (u₁v₂) - (u₂v₁)
   

Linear Algebra Example Problems - Area Of A Parallelogram
Also verify that the determinant approach to computing area yield the same answer obtained using “conventional” area computations.

Example:
Consider the parallelogram with vertices (0,0) (7,2) (5,9) (12,11)
Sketch and compute the area.

How to compute the area of a parallelogram using a determinant?

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Determinant and area of a parallelogram
Realizing that the determinant of a 2x2 matrix is equal to the area of the parallelogram defined by the column vectors of the matrix.

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