Area of a Triangle

The Determinant Formula for Area

You already know the area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by:

This expression can be written compactly as a determinant. The area $\Delta$ is:

The absolute value of this determinant gives the area, because area is always positive.

Why Collinear Points Give Zero Area

If three points lie on the same straight line, they are collinear. A triangle formed by collinear points has no area — it is degenerate. Therefore, the determinant in the area formula must be zero. …