Supplementary Topic

More on the Area of a Triangle

In the main text we said that the area of a triangle is equal to one-half of the base times height. How did we come up with this result, and how did we know that the factor of 1/2 was needed anyway?

The simplest way to try to arrive at the area of a triangle is to look at our simple - "neat" - right triangle. But ... you may ask how do we know anything about the area of a right triangle? Well ... if you recall from the section on geometry we had a kind of figure that we called a rectangle. Basically, the rectangle is what we defined as a closed figure with unequal length and width as its sides. The area of a rectangle is simply length x width. Now, imagine drawing a line from one corner of the rectangle to the opposite corner. This line is called the diagonal line and - as you can see in the figure below- it actually splits the rectangle in half. Each half of the rectangle is a right triangle because the adjacent sides of the rectangle are perpendicular to each other.

One of the right triangle regions is lined with a grid. This gridded region occupies excalty half of the area enclosed by the rectangle. Thus, the area of this gridded region - which is the area of a right triangle of one leg of length L and another leg of length W - is equal to half the area of the rectangle: Area gridded region = 1/2 L x W.

For the triangle we can consider the length L to be equivalent to the base and the width W to be equivalent to the height. Thus, the area of the right triangle is: Area = 1/2 (base) x (height).