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.
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
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.
(Note: this proof is taken from artofproblemsolving.)
Let be the set of points that belong to the polygon. Then
Note that the volume form is an exact form since , where
Substitute this in to give us
and then use Stokes’ theorem (a key theorem in vector calculus) to obtain
and is the line segment from to , i.e. is the boundary of the polygon.
Next we substitute for :
Parameterising this expression gives us
Then, by integrating this we obtain
This then yields, after further manipulation, the shoelace formula: