A lot of textbooks where I live teach students to calculate the area of a quadrilateral using its coordinates by considering it to be made up of two triangles.
We calculate the area of each triangle using the following formula:
|(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))|/2
Their individual areas are then added to calculate the final value. Won't this be a problem when calculating the area of concave quadrilaterals?
Is there any way to make sure that the area calculated using this method is indeed the correct area?
If you use the formula given as sum of products, then this is a signed area. That is, depending on the order of the three points, you may get a negative value. The formula generalizes to any polygon. This is sometimes known as the shoelace formula. As the Wikipedia article states "The area formula is valid for any non-self-intersecting (simple) polygon, which can be convex or concave". Using the general formula, the absolute value of the signed area is the area you wanted.
Let the point coordinates be $\ P_1=(x_1,y_1), P_2=(x_2,y_2), P_3=(x_3,y_3), P_4=(x_4,y_4). \ $ The area of triangle $\ P_1P_2P_4 \ $ is $\ A := x_1(y_2-y_4) + x_2(y_4-y_1) + x_4(y_1-y_2). \ $ The area of triangle $\ P_4P_2P_3 \ $ is $\ B := x_4(y_2-y_3) + x_2(y_3-y_4) + x_3(y_4-y_2). \ $ Add the two signed areas to get $\ A + B = (x_1y_2-x_2y_1) + (x_2y_3-x_3y_2) +(x_3y_4-x_4y_3) + (x_4y_1-x_1y_4) \ $ from shoelace.