Euclidean Foundations: General Triangles and Shoelace Formula (Signed Area)

Occasionally, the need arises to assign a sign to the area of a triangle, or more generally, a polygon. You may have once wanted to use the area formula $$[ABC]=[ABP]+[BCP]+[CAP]$$ for a point $P$ outside the triangle as well. (Another classic example, from physics, is calculating the work of a heat engine.)

One introduces the signed area of a (not self-intersecting) polygon $A_1A_2...A_n$ as the everyday area if $A_1,...,A_n$ are oriented counterclockwise (in the positive direction) along its perimeter, and the negative of this area if it is oriented clockwise (in the negative direction).

We'll now give another definition which is computationally more helpful and turns out to be equivalent to our first, more compact definition.

Integrals! For those who don't know yet, this is just the 'area under a function curve', or, well, in our case, we only need the area 'under a segment'. (Imagine a coordinate system.) For a segment between $(a,b)$ and $(c,d)$, this amount is defined to be $\frac{a+c}2\cdot (d-b)$.

Now for a polygon $A_1A_2...A_n$ with vertex $A_i$ at $(x_i,y_i)$, we define its area to be the sum of all the integrals of the segments $A_iA_{i+1}$ (where $A_{n+1}=A_1$). One may easily check that this gives just the same definition.

This gives us an area formula using coordinates!

$$[A_1A_2\dots A_n]_{\text{directed}}=\sum_{i=1}^n \frac{x_i+x_{i+1}}{2}\cdot (y_{i+1}-y_i)=$$ $$=\frac12 \sum_{i=1}^n (x_iy_{i+1}-x_iy_i+x_{i+1}y_{i+1}-x_{i+1}y_i)=$$ $$=\frac12 \left(\sum_{i=1}^n x_iy_{i+1}-\sum_{i=1}^n x_{i+1}y_i\right).$$

This theorem is called the shoelace formula because if we arrange the $x_i$'s and $y_i$'s in two columns and connect those who we multiply and then sum, the resulting pattern resembles a shoelace.

Specifically, for $n=3$ this becomes

$$\frac12(x_1y_2+x_2y_3+x_3y_1-x_2y_1-x_3y_2-x_1y_3),$$

or for those who know determinants,

$$\frac12 \left| \begin{array}{ccc} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{array} \right|,$$

which connects the Rule of Sarrus with the shoelace pattern.

This area formula is very helpful when dealing with coordinate geometry. Keep it in mind!

Can you convince yourself that the two signed area definitions are equivalent?

Give an algebraic criterion for points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$ to be collinear.