Calculating the Area of a Triangle with Given Vertices Using Vector Cross Product and Heron’s Formula

Discover how to efficiently find the area of a triangle when the coordinates of its vertices are known, using vector cross product and Heron’s formula.

Introduction

Knowing the area of a triangle when you have the coordinates of its vertices can be crucial for various applications in mathematics, engineering, and design. In this article, we will explore two methods to calculate the area of a triangle given its vertices:

Using the cross product of vectors Using Heron’s formula

Method 1: Using the Cross Product

Given the vertices of the triangle as (A (1, 2, 3)), (B (-2, 1, -4)), and (C (3, 4, -2)), we can start by creating two vectors from these points. If we define vector (vec{u}) as the difference between points (A) and B, and vector (vec{v}) as the difference between points (B) and C, we have:

Vector Definitions

[vec{u} (1 - (-2), 2 - 1, 3 - (-4)) (3, 1, 7)] [vec{v} (3 - (-2), 4 - 1, -2 - (-4)) (5, 3, 2)]

The cross product of these two vectors will give us a vector that is perpendicular to both (vec{u}) and (vec{v}). The magnitude of this cross product vector is twice the area of the triangle. Let's calculate this:

Cross Product Calculation

[vec{u} times vec{v} begin{vmatrix} mathbf{i} mathbf{j} mathbf{k} 3 1 7 5 3 2 end{vmatrix} mathbf{i}(1 cdot 2 - 7 cdot 3) - mathbf{j}(3 cdot 2 - 7 cdot 5) mathbf{k}(3 cdot 3 - 1 cdot 5) -19mathbf{i} 29mathbf{j} 4mathbf{k}]

The magnitude of the cross product vector is:

[|vec{u} times vec{v}| sqrt{(-19)^2 29^2 4^2} sqrt{361 841 16} sqrt{1218} approx 34.90]

The area of the triangle is half of this magnitude:

[text{Area} frac{1}{2} cdot 34.90 approx 17.45]]

Method 2: Using Heron’s Formula

For an alternative method, we can use Heron's formula to find the area of a triangle given the lengths of its sides. First, we need to calculate the side lengths:

Side Lengths

[AB sqrt{(1 - (-2))^2 (2 - 1)^2 (3 - (-4))^2} sqrt{3^2 1^2 7^2} sqrt{9 1 49} sqrt{59}] [BC sqrt{(-2 - 3)^2 (1 - 4)^2 (-4 - (-2))^2} sqrt{(-5)^2 (-3)^2 (-2)^2} sqrt{25 9 4} sqrt{38}] [AC sqrt{(1 - 3)^2 (2 - 4)^2 (3 - (-2))^2} sqrt{(-2)^2 (-2)^2 5^2} sqrt{4 4 25} sqrt{33}]

Next, we apply Heron's formula. Let (s) be the semi-perimeter:

[s frac{AB BC AC}{2} frac{sqrt{59} sqrt{38} sqrt{33}}{2}]

The area is then given by:

[text{Area} sqrt{s(s - AB)(s - BC)(s - AC)}]

Substitute the values and compute the area using a calculator for precision.

Conclusion

Both methods provide a way to calculate the area of a triangle given the coordinates of its vertices. The cross product method is faster for small triangles but becomes cumbersome for larger ones. On the other hand, Heron's formula is more straightforward and generalizable for any triangle.

Keywords: triangle area, Heron’s Formula, vector cross product, vertex coordinates