 # HERON'S FORMULA

See Also: Problem Solving with Heron's Formula

#### Introduction

Heron's formula for the area of a triangle with sides of length a, b, c is where #### Problem:

Develop a proof of Heron's Formula for the area of a triangle.

An Algebraic Proof

This demonstration of Heron's formula is straightforward and elementary. Working through it with students can provide fruitful ideas of strategy, symmetry, planning, and observation. We now switch to consider some problems and investigations for which Heron's formula is useful.

A Trigonometric Shortcut

In the algebraic proof, a value p was computed in terms of a, b, and c. Consider the Law of Cosines and examine p as the cosine of the angle between a and c.

Considered by some mathematics people to be a "more elegant approach." What do you think?

A Geometric Proof

Heron would have used geometric ideas to develop and prove this formula.  In the history of mathematics course a geometric proof is usually considered that is similar to Heron's methods.

A Modern (and elegant) Geometric Proof

A Demonstration with Complex Numbers

Return to the EMAT 4400/6600 Page