EXPLORATIONS AND PROBLEM SOLVING WITH HERON'S FORMULA
AND ALTERNATIVE PROOFS
Heron's formula for the area of a triangle with sides of length a, b, c is
Historical Note: Heron of Alexandria (sometimes called Hero) lived from approximately 10 AD to approximately 75 AD. A biography of him can be found on the MacTutor History of Mathematics Site.
In this presentation, I hope to accomplish two general things. First, I hope to show some ways to derive Heron's formula and along the way understand how it is a viable alternative for giving the area of a triangle. Second I hope to show some examples of interesting problems for which using Heron's formula might be useful. By "interesting" of course I am saying they are interesting for me. I hope they also do the trick of others.
For those who want to tune out discussions of the derivations, here are a couple of problems you can think about while the rest of us are attending to the derivations:
It is unfortunate that this topic has essentially disappeared from school curriculum today. Calculation, given available calculations and computers, can no longer be a reason for avoiding the formula. After discussion of derivations and proofs of Heron's formula, I hope to show some interesting and challenging problems using Heron's formula.
Ideas about Derivation and Proof of Heron's Formula
Whether or not one would pose the demonstration or proof of Heron's formula for a particular class would depend on the class. Initially, exploration with Heron's formula could involve computing areas using the formula and making comparison's of the results -- much as we pose analogous exercises in a meaningful way with the Pythagorean theorem long before a proof or demonstration is fully understood.
There are several derivations of Heron's Theorem in the literature. Unfortunately, the most well-known and most often cited are complicated or tedious or both.
Usually, a derivation starts with some geometric idea of an area formula for a triangle involving some measures other than just the sides of the triangle. The most usual, of course is that the area is one half a base times the altitude to the base.
This is the approach for the most common algebraic derivation that will be outlined below. Here getting Heron's formula depends on getting an expression for one of the altitudes in terms of the lengths of the sides.
It is also the beginning of some derivations using trigonometry. I will outline one of them later.
An alternative start, however, is to consider the triangle with its incircle and in-radius of r. The area of the triangle is the sum of the area measures of the three subtriangles IAC, IBC, and IAB. Each of those subtriangles has an altitude of r and deriving Heron's Formula depends on getting an expression for r in terms of the lengths of the sides.
I will show 4 derivations starting with this perspective. The geometric proof most common in the literature shows a geometric approach thought to be similar to what Heron used. It develops the formula from this perspective.
Lemma: If a circle is tangent to the two sides of an angle, then the distances from the vertex to the points of tangency on each side are equal and the center is on the angle bisector.