Using the determinant of a matrix to calculate the area of a triangle

This formula gives the area of the triangle formed by the vertices

(x₁,y₁), (x₂,y₂), and (x₃,y₃)

The determinant of this matrix is

D = x₁y₂ + x₂y₃ + x₃y₁ - x₁y₃ - x₂y₁ - x₃y₂

This can be partially factored as

D = (x₁-x₃)(y₂-y₃) - (x₂-x₃)(y₁-y₃)

Now move the vertex (x₃,y₃) to the origin by subtracting it from each of the vertices.  So now our triangle becomes

(x₁-x₃,y₁-y₃), (x₂-x₃,y₂-y₃), and (0,0)

Naturally, this triangle has the same area as the original triangle; All I've done is move it.  To simplify the discussion, let

a = (x₁-x₃),
b = (y₁-y₃),
c = (x₂-x₃), and
d = (y₂-y₃)

Now the triangle in question has vertices (a,b), (c,d), and (0,0).  Substituting these variables into the equation, above, for the determinant of the original matrix, we see it is

D = ad - bc

The area of the triangle, then, using the determinant formula is

A = (1/2)(ad - bc)

Well, not quite.  The area can't be negative, so we have to take the absolute value.

A = (1/2)|ad-bc|

Let's review.  Here is what we know so far

(1) The determinant of the 3x3 matrix, above, is equal to ad-bc, after we set a, b, c, and d appropriately.

(2) The area of triangle (a,b), (c,d), (0,0) is the same as the area of triangle
(x₁,y₁), (x₂,y₂), and (x₃,y₃)

Now, to finish the explanation of why half the determinant of that 3x3 matrix is equal to the triangle's area, we just need to explain why

A = (1/2)|ad-bc|

For this explanation, click here: Triangle Area using Vectors

Related Pages in this website

Polygon Area using Determinant

Introduction to Matrices

Matrix Definitions

Cramer's Rule

Heron's formula for the area of a triangle, if all you know is the lengths of its sides.

Algorithm to determine whether a point is inside a triangle.

The webmaster and author of the Math Help site is Graeme McRae.
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