An [[area]] formula for simple [[polygon|polygons]] with vertices $(x_0,y_0), (x_1,y_1) \ldots (x_{n-1},y_{n-1})$ This form sums signed areas of triangles to form the total area $A=\frac{1}{2} \sum_{i=0}^{n-1}\left(x_i y_{i+1}-x_{i+1} y_i\right) \quad \text{note: }x_n=x_0,\ y_n=y_0$ each difference is a determinant of a matrix that allows rewriting $\begin{aligned} A & =\frac{1}{2} \sum_{i=1}^n\left(x_i y_{i+1}-x_{i+1} y_i\right)=\frac{1}{2} \sum_{i=1}^n\left|\begin{array}{ll}x_i & x_{i+1} \\ y_i & y_{i+1}\end{array}\right|=\frac{1}{2} \sum_{i=1}^n\left|\begin{array}{cc}x_i & y_i \\ x_{i+1} & y_{i+1}\end{array}\right| \\ & =\frac{1}{2}\left(x_1 y_2-x_2 y_1+x_2 y_3-x_3 y_2+\cdots+x_n y_1-x_1 y_n\right)\end{aligned}$ Which leads to a scheme which chains through the sequence of vertices in a shoelace pattern.