MATHS BITE: Shoelace Theorem

The Shoelace theorem is a useful formula for finding the area of a polygon when we know the coordinates of its vertices. The formula was described by Meister in 1769, and then by Gauss in 1795.

Formula

Let’s suppose that a polygon P has vertices (a1, b1), (a2, b2), …, (an, bn), in clockwise order. Then the area of P is given by

$\dfrac{1}{2} |(a_1b_2 + a_2b_3 + \cdots + a_nb_1) - (b_1a_2 + b_2a_3 + \cdots + b_na_1)|$

The name of this theorem comes from the fact that if you were to list the coordinates in a column and mark the pairs to be multiplied, then the image looks like laced-up shoes.

Proof

(Note: this proof is taken from artofproblemsolving.)

Let $\Omega$ be the set of points that belong to the polygon. Then

$A=\int_{\Omega}\alpha,$

where $\alpha=dx\wedge dy$ .

Note that the volume form $\alpha$ is an exact form since $d\omega=\alpha$ , where

$\omega=\frac{x\,dy}{2}-\frac{y\,dx}{2}.\label{omega}$

Substitute this in to give us

$\int_{\Omega}\alpha=\int_{\Omega}d\omega.$

and then use Stokes’ theorem (a key theorem in vector calculus) to obtain

$\int_{\Omega}d\omega=\int_{\partial\Omega}\omega.$

where

$\partial \Omega=\bigcup A(i)$

and $A(i)$ is the line segment from $(x_i,y_i)$ to $(x_{i+1},y_{i+1})$ , i.e. Screen Shot 2017-08-04 at 12.05.20 PM.png is the boundary of the polygon.