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.
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 Green 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: