where is the semiperimeter of the triangle:
Heron's formula can also be written as:
The formula is credited to Heron 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, and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that it predates the reference given in the work.
A formula equivalent to Heron's namely:
A modern proof, which uses algebra and trigonometry and is quite unlike the one provided by Heron, follows. Let a, b, c be the sides of the triangle and A, B, C the angles opposite those sides. We have
by the law of cosines. From this we get the algebraic statement:
The altitude of the triangle on base a has length b sin(C), and it follows
The difference of two squares factorization was used in two different steps.
Proof using the Pythagorean theoremHeron's original proof made use of cyclic quadrilaterals, while other arguments appeal to trigonometry as above, or to the incenter and one excircle of the triangle . The following argument reduces Heron's formula directly to the Pythagorean theorem using only calculus means.
In the form 4A 2 = 4s(s − a)(s − b)(s − c), Heron's formula reduces on the left to (ch)2, or
using b 2 − d 2 = h 2 by the Pythagorean theorem, and on the right to
via the identity (p + q) 2 − (p − q) 2 = 4pq. It therefore suffices to show
The former follows immediately by reflection (a + b + c)/2 for s and simplifying. Doing this for the latter reduces s(s − a)(s − b)(s − c) only as far as (b 2 + c 2 −a 2)/2. But if we replace b 2 by d 2 + h 2 and a 2 by (c − d) 2 + h 2, both by Pythagoras, simplification then produces cd as required.
The parentheses in the above formula are required in order to prevent numerical instability in the evaluation.
Heron's formula is a special case of Brahmagupta's formula for the area of a cyclic quadrilateral; both of which are special cases of Bretschneider's formula for the area of a quadrilateral. In both cases Heron's formula is obtained 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.
Heron-looking formula for tetrahedrons
- Heath, Thomas L. (1921). A History of Greek Mathematics (Vol II). Oxford University Press. pp. 321–323.
- MathWorld entry on Heron's Formula
- A Proof of the Pythagorean Theorem From Heron's Formula at cut-the-knot
- Interactive applet and area calculator using Heron's Formula
- Implementations of Heron's formula in various programming languages
- J.H. Conway discussion on Heron's Formula
- Kevin Brown's simplification of Heron's Pythagorean argument
- Step by Step Proof of Heron's Formula.
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