An Alternate Method to Find Area of Triangle from its Vertices alone in a Plane

Abstract

Since the time of ancient Greeks, mathematicians have been interested in finding the areas of basic plane regions. From among the most basic plane regions are triangles.  Triangular shapes are used in different areas of engineering, especially in the design and analysis of trusses and in finding moments of inertia in mechanics and to find triangulation and trilateration in surveying. The centroid of a triangular region can be also expressed in terms of the coordinates of vertices of triangles.  Literatures have shown that there are various methods of finding an area of a triangle. From among these are using the length of the base and height of a triangle, using the length of two sides and the sine of included angle or using Heron`s formula which uses the length of all sides of the triangle. We can also find the area of a triangle if the coordinates of the vertices are given using either vectors or determinants.  However, vectors and determinants are concepts of higher mathematics. In this article, we present a method that enables us to find the area of a triangle without the knowledge of vectors or determinants, provided that the coordinates of the vertices of a triangle were given. We use analytic approach to derive the formula. The method use only elementary arithmetic. To this end a procedure were designed and the method were checked for accuracy using different examples. Finally, a theorem was formulated and proved.