1. Heron's formula In geometry, Heron's (or Hero's) formula, named after Heron of Alexandria, [1] states that the area T of a triangle whose sides have lengths a, b, and c isgeometryHeron of Alexandria [1]areatriangle where s is the semiperimeter of the triangle:semiperimeter Heron's formula can also be written as:

2. History The formula is credited to Heron (or Hero) of Alexandria, and a proof can be found in his book, Metrica, written c. A.D. 60. It has been suggested that Archimedes knew the formula over two centuries earlier, and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work. [2]Heron (or Hero) of AlexandriaArchimedes [2] A formula equivalent to Heron's namely:, where was discovered by the Chinese independently of the Greeks. It was published in Shushu Jiuzhang (“Mathematical Treatise in Nine Sections”), written by Qin Jiushao and published in A.D. 1247.Mathematical Treatise in Nine SectionsQin Jiushao

3. Proof using the Pythagorean theorem Heron's original proof made use of cyclic quadrilaterals, while other arguments appeal to trigonometry as above, or to the incenter and oneexcircle of the triangle [2]. The following argument reduces Heron's formula directly to the Pythagorean theorem using only elementary means.cyclic quadrilateralstrigonometryincenterexcircle[2]Pythagorean theorem In the form 4T 2 = 4s(s − a)(s − b)(s − c), Heron's formula reduces on the left to (ch) 2, or which is the same as using b 2 − d 2 = h 2 by the Pythagorean theorem, and on the right toPythagorean theorem via the identity (p + q) 2 − (p − q) 2 = 4pq. It therefore suffices to show and Then expanding the former you get the following: and that reduces to by substituting 2s = (a + b + c) and simplifying. Submitting for s the latter s(s − a) − (s − b)(s − c) reduces only as far as (b 2 + c 2 − a 2 )/2. But if we replace b 2 byd 2 + h 2 and a 2 by (c − d) 2 + h 2, both by Pythagoras, simplification then produces cd as required.

4. HERONS FORMULA NUMERICAL UNSTABLE