On Heron’s Formula for the Area of a Plane Triangle

Abstract

Summary This paper provides a review of proofs of Heron’s formula for the area of a plane triangle. We groups proofs together into analogy/experimental evidence, geometry (incircle radius multiplied by semi-perimeter), algebra (equivalence with the Pythagorean Theorem), trigonometric proofs (often via the cosine rule), and specialization from a more general theorem. Lastly, we discuss a neglected paper by Baker of 1885 that lists 110 distinct formulae for the area of a plane triangle and relate some of these formulae to those used in proving Heron’s formula.