Heron's Formula Heron's Formula In geometry, Heron's (or Hero's) formula, named after Heron of Alexandria, states that the area T of a triangle whose sides have lengths a, b, and c is where s is the semiperimeter of the triangle: Heron's formula is distinguished from other formulas for the area of a triangle, such as half the base times the height or half the modulus of a cross product of two sides, by requiring no arbitrary choice of side as base or vertex as origin. 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. A formula equivalent to Heron's namely :
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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. Proof :- A modern proof, which uses algebra and is quite unlike the one provided by Heron (in his book Metrica), follows. Let a, b, c be the sides of the triangle and A, B, C the angles opposite those sides. Heron's formula as given above is numerically unstable for triangles with a very small angle. A stable alternative involves arranging the lengths of the sides so that and computing The brackets in the above formula are required in order to prevent numerical instability in the evaluation. Generalizations : Heron's formula is a special case of Brahmagupta's formula for the area of a cyclic quadrilateral. Heron's formula and Brahmagupta's formula are both special cases of Bretschneider's formula for the area of a quadrilateral. Heron's formula can be obtained from Brahmagupta's formula or Bretschneider's formula by setting one of the sides of the quadrilateral to zero. Heron's formula is also a special case of the formula of the area of the trapezoid based only on its sides. Heron's formula is obtained by setting the smaller parallel side to zero. Expressing Heron's formula with a Cayley–Menger determinant in terms of the squares of the distances between the three given vertices, illustrates its similarity to Tartaglia's formula for the volume of a three-simplex.
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Another generalization of Heron's formula to pentagons and hexagons inscribed in a circle was discovered by David P. Robbins. Herons formula is based upon the properties of cyclic quadrilaterals and right Triangles. The heron's formula is named after the mathematician Heron of Alexandria. The herons formula is derived from the other formulas of area of a triangle. Suppose that the each side of a triangle is 6 inch long then area of the triangle using herons formula can be calculated as:
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