Hero's Formula, also known as Heron's Formula, is a method for calculating the area of a triangle when the lengths of all three sides are known. This formula is named after Hero of Alexandria, a Greek mathematician, who is credited with its discovery, although some sources suggest it might have been known earlier.
The formula is expressed as: \[ A = \sqrt{s(s-a)(s-b)(s-c)} \] where:
While Hero of Alexandria (c. 10–70 AD) is often credited with the formula, evidence suggests that the principle behind it was known to the ancient Egyptians, as similar methods appear in the Rhind Papyrus. However, Hero's work, particularly his book Metrica, which was rediscovered in 1896, provides the earliest written record of this formula in the Western world. Hero's contributions were not limited to this formula; he was also known for his work in mechanics, pneumatics, and mathematics, significantly influencing the development of these fields.
Hero's Formula is particularly useful in scenarios where:
Several proofs exist for Hero's Formula, with the most common ones involving algebraic manipulation of the triangle's properties or geometric constructions. One intuitive proof involves using the Law of Cosines to find the cosine of an angle and then applying trigonometric identities to derive the area formula.
The formula has been extended to various other geometric shapes: