area of polygon

Let the vertices of a () polygon be $(x_{1},\\,y_{1}),\\,(x_{2},\\,y_{2}),\\,\\ldots,\\,(x_{n},\\,y_{n})$ , enumerated in 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).

References

1 E. Lindelöf: Johdatus korkeampaan analyysiin. Neljäs painos. Werner Söderström Osakeyhtiö, Porvoo and Helsinki (1956).