A Detailed Derivation of the Heron’s Formula
Heron’s Formula can be used to find the area of a triangle given the lengths of the three sides. A triangle with side lengths , , and , its area can be calculated using the Heron’s formula
is the semiperimeter (half the perimeter) of the triangle.
In this post, I will provide a detailed derivation of this formula.
The area of a triangle is half the product of its base and its altitude. In the figure below, is the altitude of triangle . If the length of the altitude is not given, and an angle measure is given, we can use Trigonometry to calculate the altitude.
In the figure above, the altitude forms right triangle . We know that
= (length of opposite side)/(length of hypotenuse)
Simplifying, we have which is equivalent to
Since the base of triangle is and its altitude is , its area is given by the formula
Now we apply the preceding formula, the Cosine Law and the Pythagorean identity to derive the Heron’s formula.
Using the Pythagorean identity, and manipulating algebraically
By the Cosine Law, in a triangle with side lengths , , and
Calculating for , we have
Substituting the preceding equation to (2), we have
Getting the square root of both sides, we have
Using (1) and (3), we calculate the area of the triangle ,
Now, if we let be the semiperimeter (half the perimeter) of triangle , then
Also, and .
Substituting the expressions with s to (4), we have
which is equivalent to
Simplifying, we have , the Heron’s formula.