Suppose the polygon has vertices , , ... , , listed in clockwise order. Then the area of is
The Shoelace Theorem gets its name because if one lists the coordinates in a column, and marks the pairs of coordinates to be multiplied, the resulting image looks like laced-up shoes.
Lemma 1: The area of a triangle with coordinates , , and is .
Let's translate to the origin so that the other two points are now and .
We will proceed with induction. We start by proving it is true for a triangle: Let the triangle have coordinates and . To simplify calculations let's translate to the origin.
Let be the set of points belonging to the polygon. We have that where . The volume form is an exact form since , where Using this substitution, we have Next, we use the theorem of Stokes to obtain We can write , where is the line segment from to . With this notation, we may write If we substitute for , we obtain If we parameterize, we get Performing the integration, we get More algebra yields the result
In right triangle , we have , , and . Medians and are drawn to sides and , respectively. and intersect at point . Find the area of .
A good explanation and exploration into why the theorem works by James Tanton:  AOPS