Area enclosed by a polygon

Question

I did some work in the area of mensuration and came across an interesting concept/formula.

The formula states that

For a polygon having vertices $(x_1,y_1),(x_2,y_2),(x_3,y_3),\dots(x_n,y_n)$, the area enclosed by the polygon is half the sum of  the $2\times2$ determinants of consecutive points on its vertices as we proceed in an anticlockwise loop from an initial vertex back to the same vertex.

It provides a way of finding the area of a closed $n$-sided polygon(regular or not) using a determinant loop taken about its vertices.

See the image here

See my blog about the subject here

I would like to know

1. the correctness of the formula

2. If there exists a similar formula in mathematics.

Answer 1

You're describing the famous shoelace formula, also known as Gauss's area formula: https://en.wikipedia.org/wiki/Shoelace_formula

Answer 2

You’re essentially decomposing the polygon into triangles and adding up their areas. Those determinants give the area of the parallelogram spanned by each pair of vectors, so the area of the triangle with those two sides is half that.