The area of a parallelogram with given vertices in rectangular coordinates can be calculated using the vector cross product. The area of a parallelogram is equal to the product of its base and height. Using vector values derived from the vertices, the product of a parallelogram's base and height is equal to the cross product of two of its adjacent sides. Calculate the area of a parallelogram by finding the vector values of its sides and evaluating the cross product.
Find the vector values of two adjacent sides of the parallelogram by subtracting the x and y values of the two vertices that form the side. For example, to find length DC of parallelogram ABCD with vertices A (0, -1), B (3, 0), C (5, 2) and D (2, 1), subtract (2, 1) from (5, 2) to get (5 - 2, 2 - 1) or (3, 1). To find length AD, subtract (2, 1) from (0, -1) to get (-2, -2).
Write a matrix of two rows by three columns. Fill in the first row with the vector values of one side of the parallelogram (the x value in the first column and the y value in the second) and write zero in the third column. Fill in the values of the second row with the vector values of the other side and zero in the third column. In the above example, write a matrix with the values {{3 1 0}, {-2 -2 0}}.
Find the x-value of the cross product of the two vectors by blocking out the first column of the 2 x 3 matrix and calculating the determinant of the resulting 2 x 2 matrix. The determinant of a 2 x 2 matrix {{a b}, {c d}} is equal to ad - bc. In the above example, the x-value of the cross product is the determinant of the matrix {{1 0}, {-2 0}}, which is equal to 0.
Find the y-value and z-value of the cross product by blocking out the second and third columns of the matrix, respectively, and calculating the determinant of the resulting 2 x 2 matrices. The y-value of the cross product is equal to the determinant of the matrix {{3 0}, {-2 0}}, which is equal to zero. The z-value of the cross product is equal to the determinant of the matrix {{3 1}, {-2 -2}}, which is equal to -4.
Find the area of the parallelogram by calculating the magnitude of the cross product <x,y,z> using the formula √(x^2 + y^2 + z^2). In the above example, the magnitude of the cross product vector <0,0,-4> is equal to √(0^2 + 0^2 + (-4)^2), which is equal to 4.
When Is This Useful?
Finding the area of a parallelogram can be useful in many areas of study including mathematics, physics and biology.
Mathematics
Mathematics studies are probably the most obvious use of finding the area of a parallelogram. Knowing how to find the area of parallelogram in coordinate geometry is often one of the first things you'll do before moving on to more complex shapes. This can also introduce you to more complex graphing and vector/vertices based mathematics that you'll see in upper level math classes, geometry, coordinate geometry, calculus and more.
Physics
Physics and mathematics go hand in hand and that's certainly true with vertices. Knowing how to find the area of a parallelogram this way can extend to finding other areas as well like a problem that requires you to find the area of the triangle with vertices in a physics problem on velocity or electromagnetic force, for example. The same concept of coordinate geometry and calculating the area can apply to a number of physics problems.
References
About the Author
Karl Wallulis has been writing since 2010. He has written for the Guide to Online Schools website, covering academic and professional topics for young adults looking at higher-education opportunities. Wallulis holds a Bachelor of Arts in psychology from Whitman College.
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