Heron's formula

In geometry, Heron's formula (sometimes called Hero's formula), named after Hero of Alexandria, gives the area of a triangle by requiring no arbitrary choice of side as base or vertex as origin, contrary to other formulas for the area of a triangle, such as half the base times the height or half the norm of a cross product of two sides.

Heron's formula states that the area of a triangle whose sides have lengths a, b, and c is

where s is the semiperimeter of the triangle; that is,

Heron's formula can also be written as

Let △ABC be the triangle with sides a = 4, b = 13 and c = 15. The semiperimeter is s = 1/2(a + b + c) = 1/2(4 + 13 + 15) = 16, and the area is

In this example, the side lengths and area are all integers, making it a Heronian triangle. However, Heron's formula works equally well in cases where one or all of these numbers is not an integer.

The formula is credited to Heron (or Hero) of Alexandria, and a proof can be found in his book, Metrica, written c. CE 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.

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