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Shoelace Theorem

by Eugene Chen of the Amador Valley Math Team

We derive a formula for the polygon with vertices (x_1,y_1),(x_2,y_2),\dots,(x_n,y_n):

(a) Show that the signed area of the triangle with vertices (0,0), (x_1,0), (x_2,y_2) is \frac {x_1y_2}{2}.

(b) Explain why the area of a polygon stays the same after a shear is performed.

(c) Compute the signed area of the triangle with vertices (0,0),(x_1,y_1),(x_2,y_2).

(d) The signed area of the polygon (x_1,y_1),(x_2,y_2),\dots,(x_n,y_n) can be written as the sum of the signed area of n triangles, each of which has a vertex at (0,0) and two vertices in common with the polygon. Explain why.

(e) Find a formula for the signed area of the polygon with vertices (x_1,y_1),(x_2,y_2),\dots,(x_n,y_n).

(a-hint) Consider when x_1,y_2 are positive, negative, etc.
(b-hint) Consider a square in the plane. What shape does it turn into after a shear? What is its area?
(c-hint) Perform a shear.
(d-hint) Show that the statement is true for triangles, then use induction.
(e-hint) Use (d) - add the areas of the triangles.

The Shoelace Theorem states the following:

Theorem: Suppose the polygon P has vertices (x_1, y_1), (x_2, y_2), \dots , (x_n, y_n), listed in clockwise or counter-clockwise order. Then the area of P is
\dfrac{1}{2} |(x_1y_2 + x_2y_3 + \cdots + x_ny_1) - (x_2y_1 + x_3y_2 + \cdots + x_1y_n)|
It is easier to remember when considering it in a different form, which it gets its name from:
\left [ \begin{tabular}{rl} x_1 & y_1 \\ x_2 & y_2 \\ \vdots & \vdots \\ x_n & y_n \\ x_1 & y_1\end{tabul...
Determine the sum of the products going south-east (that is, x_1y_2 + x_2y_3 + \dots x_ny_1) and the sum of the products going northeast (that is, x_2y_1 + x_3y_2 + \dots + x_1y_n). Half the absolute value of their difference is the area of the polygon. If you draw the diagonals, you'll see why it's named the Shoelace Theorem.

It's much easier to show with an example:

Example 1: Find the area of the pentagon in the diagram below.

dot((2,-4)); dot((5,2)); dot((3,7)); dot((-4,5)); dot((-6,-2)); draw((2,-4)--(5,2)--(3,7)--(-4,5)--(-6,-2)--cycle); label(&qu...

Use the Shoelace Theorem:

Image

The picture above shows the drawn-in diagonals. We see that the area is \frac {1}{2} \left | (2)(2) + (5)(7) + (3)(5) + ( - 4)( - 2) + ( - 6)( - 4) - ((5)( - 4) + (3)(2) + ( - 4)(7) + ( - 6)(5) + (.... This simplifies to \frac {1}{2} \left | 86 - ( - 76) \right | = \boxed{81}, which is the area of the pentagon.

Exercise: Write a few random coordinates, graph them, and find the area of the polygon with those vertices (make sure the vertices are in clockwise or counter-clockwise order when finding the area).
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