Using Determinant to find the Area of a Parallelogram
Related Pages
Determinant of a 2×2 Matrix
Inverse of a 2×2 Matrix
Matrices
More Lessons for Grade 9
These lessons, with videos, examples and step-by-step solutions, help Algebra students learn how to use the determinant to find the area of a parallelogram. You can use the determinant of a matrix formed by the vectors representing the adjacent sides of a parallelogram to find its area.
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?
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Represent the Sides as Vectors:
Let the two adjacent sides of the parallelogram be represented by vectors u = (u1, u2) and v = (v1, v2). -
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:
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Calculate the Determinant:
The determinant of matrix M is: det(M) = (u1v2) - (u2v1) -
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?
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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