Area of a triangle in the complex plane

Posted on by John

I recently ran across an elegant equation for the area of a triangle in the complex plane with vertices at z1, z2, and z3. [1].

A(z_1, z_2, z_3) = \frac{i}{4} \, \left| \begin{matrix} z_1 & \bar{z}_1 & 1 \\ z_2 & \bar{z}_2 & 1 \\ z_3 & \bar{z}_3 & 1 \\ \end{matrix} \right|

This formula gives the signed area: the area is positive if the points are given in countclockwise order and negative otherwise.

I’ll illustrate the formula with a little Python code. Let’s generate a random triangle.

    import numpy as np

    np.random.seed(20221204)
    r = 100*np.random.random(6)
    z1 = r[0] + 1j*r[1]
    z2 = r[2] + 1j*r[3]
    z3 = r[4] + 1j*r[5]

Here’s what our triangle looks like plotted.

Now let’s calculate the area using the formula above and using Heron’s formula.

    
    def area_det(z1, z2, z3):
        det = 0
        det += z2*z3.conjugate() - z3*z2.conjugate()
        det -= z1*z3.conjugate() - z3*z1.conjugate()
        det += z1*z2.conjugate() - z2*z1.conjugate()
        return 0.25j*det
    
    def area_heron(z1, z2, z3):
        a = abs(z1-z2)
        b = abs(z2-z3)
        c = abs(z3-z1)
        s = 0.5*(a + b + c)
        return np.sqrt(s*(s-a)*(s-b)*(s-c))
        
    print(area_heron(z1, z2, z3))
    print(area_det(z1, z2, z3))

This prints -209.728 and 209.728. The determinate gives a negative area because it was given the points in clockwise order.

[1] Philip J. Davis. Triangle Formulas in the Complex Plane. Mathematics of Computation. January 1964.

2 thoughts on “Area of a triangle in the complex plane”

  1. Mel

    This technique is easy to teach to high school geometry students that have no knowledge of complex numbers or determinants.

    A little analytical geometry is all.
    Fun fact for them.

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