Let the vertices of a (planar) polygon be $(x_1,\,y_1),\,(x_2,\,y_2),\,\ldots,\,(x_n,\,y_n)$ enumerated in order when gone round the polygon anticlockwise. The area of the polygon is equal to $$ \frac{1}{2}\left(\left|\begin{matrix}x_1&x_2\\y_1&y_2\end{matrix}\right| +\left|\begin{matrix}x_2&x_3\\y_2&y_3\end{matrix}\right|+\ldots +\left|\begin{matrix}x_{n-1}&x_n\\y_{n-1}&y_n\end{matrix}\right|+ \left|\begin{matrix}x_n&x_1\\y_n&y_1\end{matrix}\right|\right). $$

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