Analytic Geometry

One of the most frequent activities in geometry is determining the area of a polygon such as a triangle or square. By using coordinates to represent the vertices, the areas of any polygon can be determined. The area of triangle OPQ, where O lies at (0,0), P at (a,b), and Q at (c,d), is found by first calculating the area of the entire rectangle and subtracting the areas of the three right triangles. Thus the area of the triangle formed by points OPQ is = da — (dc/2) — (ab/2) — [(d — b)(a — c)]/2. Through the use of a determinant, it can be shown that the area of this triangle is:

This specific case was made easier by the fact that one of the points used for a vertex was the origin.

The general equation for the area of a triangle defined by coordinates is represented by the previous equation.

In a similar manner, the area for any other polygon can be determined if the coordinates of its points are known.