"Published in Newark, California, USA"
Given the triangle below:
 |
| Photo by Math Principles in Everyday Life |
Prove that the area of triangle is
Solution:
Since the given triangle has no altitude, we have to assign it anywhere from the vertices of a given triangle. From point B, draw a line that is perpendicular to b and label this as h as the altitude.
 |
| Photo by Math Principles in Everyday Life |
We know that the area of triangle is
but
Square the both sides of the equation
but
From Cosine Law,
Square the above equation
Therefore,
You notice that we can factor the above equation by the difference of two squares
Arrange the above equation and you notice that we can factor again by the difference of two squares
If a + b + c is the perimeter of a triangle, then s is the semi-perimeter of a triangle. Substitute
to the above equation. Therefore,
The above formula is called Heron's Formula.