As a summary, given the vertices of the polygon [math]P_1(x_1,y_1)[/math], [math]P_2(x_2,y_2)[/math] ,[math]\cdots[/math], [math]P_n(x_n,y_n)[/math], the area of a polygon is given by this formula:[br][center][math]\text{Area of a Polygon}=\frac{1}{2}\begin{vmatrix}x_1 & x_2 & x_3 & \cdots & x_n & x_1 \\y_1 & y_2 & y_3 & \cdots & y_n & y_1 \end{vmatrix}[/math][/center]where the products taken in the direction[math]\searrow[/math]are given [color=#38761d][b]positive [/b][/color]signs, [br]and products taken in the direction[math]\nearrow[/math]are given [b][color=#cc0000]negative [/color][/b]signs.[br][br]E.g. for a polygon with three vertices [math]P_1(x_1,y_1),P_2(x_2,y_2),P_3(x_3,y_3)[/math], the area of the polygon is [center][math]\begin{align}\text{Area of a Polygon}&=\frac{1}{2}\begin{vmatrix}x_1 & x_2 & x_3 & x_1 \\y_1 & y_2 & y_3 & y_1 \end{vmatrix}\\ &= \frac{1}{2}x_1 y_2+x_2 y_3 + x_3 y_1 - \left(x_2 y_1 + x_3 y_2 + x_1 y_3\right)\end{align}[/math][/center][b]NOTE[/b]: this formula only works fine when the vertices are considered in a [b][u]counterclockwise direction[/u][/b].