Let a simple polygon on the plane be defined by its vertices . Then, the area of the polygon is . When the coordinates are written out on a matrix and the multiplications are drawn with different lines
This can also be written in form of a summation or in terms of determinants as which is useful in the variant of the Shoelace theorem. Note here that and .
The formula may also be considered a special case of Green's Theorem
where and so .
Claim 1: The area of a triangle with coordinates , , and is .
Proof of claim 1:
Writing the coordinates in 3D and translating so that we get the new coordinates , , and . Now if we let and then by definition of the cross product .
We will proceed with induction.
By claim 1, the shoelace theorem holds for any triangle. We will show that if it is true for some polygon then it is also true for .
We cut into two polygons, and . Let the coordinates of point be . Then, applying the shoelace theorem on and we get
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
This is a very nice approach that directly helps in understanding the sum as terms which are areas of trapezoids.
The proof is in this book: https://cses.fi/book/book.pdf#page=281
(The only thing that needs to be slightly modified is that one must shift the entire polygon up by k, until all the y coordinates are positive, but this term gets canceled in the resulting sum.)
In right triangle , we have , , and . Medians and are drawn to sides and , respectively. and intersect at point . Find the area of .
Observe that is the area of a triangle with vertices and is the volume of a tetrahedron with vertices . Does a similar formula hold for Dimensional triangles for any ? If so how can we use this to derive the D Shoelace Formula?
A good explanation and exploration into why the theorem works by James Tanton: 
Nice geometric approach and discussion for proving the 3D Shoelace Theorem by Nicholas Patrick and Nadya Pramita: 
Nice integral approach for proving the 3D Shoelace Theorem (ignoring sign of volume) by @george2079:  AOPS