Correct way to find the area of a concave quadrilateral in co-ordinate geometry using triangles. 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?
 A: 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.
A: You can surely use that method but it would be quite "bashy". Especially if you do it by hand. A better tool for this case (or any polygon) would be Gauss's Area Theorem, AKA shoelace theorem. I think this page provides a better explanation if shoelace theorem than Wikipedia.
https://artofproblemsolving.com/wiki/index.php?title=Shoelace_Theorem
