Geometry of plane areas and 2 X 2 determinants

This post gives an insight into how to compute the area of closed, regular and irregular plane areas such as rectangle, triangles, squares, trapezoidal shapes, polygons and etc. both regular and irregular.

We show a simple link between 2 X 2 determinants and plane areas, using a principle I like to call the DETERMINANT LOOP RULE. Here is a simple statement of the rule:

For a polygon having vertices (x1,y1), (x2,y2), (x3,y3),........(xn,yn), the area enclosed by the polygon is half the sum of  the 2X2 determinants of consecutive points on its vertices as we proceed in an anticlockwise loop from an initial vertex back to the same vertex.

For any 2 consecutive vertices, we form the determinant like this:

Constructing the determinant from 2 points.

To compute the area of a 3 sided plane shape, we have:

Area of a triangle with the determinant loop rule

In general, a mathematical expression of the rule is given below as:

Mathematical statement of the determinant loop rule

For example, to compute the area of the triangle whose vertices are at: (15,16), (5,5), and (20,0), we have:

Area of a triangle from first principles with the determinant loop rule

And of course, some very queer shapes can be very quickly dealt with area-wise, e.g. the very irregular shape below:

Computing the area of an irregular figure with the determinant loop rule

Pretty cool, huh?


In fact, this rule is so general that it can be applied to curves, to generate the area under a curve and so get the link between integral calculus and determinants. We could take neighboring points on a curve and let dx(the horizontal distance between them) tend to zero. We could the apply the determinant loop rule to these points and the x axis. e.g.

Integral calculus and the determinant loop rule.

As expected, the rule as stated applies directly to this diagram and computes the integral of the shape above between x1 and x2. Accuracy increase as dx tends to zero.


Further research shows that this rule is a direct link between 2X2 determinants and integral calculus.

Infact for a given dx, there is a one to one mapping between accuracies gotten here and accuracies gotten with the trapezoidal rule of integration.

It has been nice sharing knowledge  fellows. Comments would be really nice and appreciated.