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On Taking Different Approaches: Finding the Area of a Triangle
Different views reveal different truths
There are many general maxims and rules to live by in mathematics and one of the more innocent-looking is the fact that it is good to have multiple different approaches to the same thing. Different ways of doing something can shed very different light on it and will often generalise in very different ways. Moreover, this applies to even the simplest of pieces of mathematics. To see what I mean lets consider a calculation so basic that it is routinely done by school children the world over: finding the area of a triangle.
As is traditional, throughout this piece we will consider a triangle whose sides we will label a, b and c whilst we give the name A to the angle opposite the side of length a and similarly for B and C.
The Classic Approach
As everyone knows by far the simplest and most basic approach to finding a triangle’s area is when we know its height and its width. As you probably know we have that
Area = (height × width) / 2
This comes from little more than the observation that splitting a rectangle in two by adjoining opposite diagonals halves its area.
