Heron's formula-白红宇

Heron's formula

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发布时间：2019-04-28

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This article is about calculating the area of a triangle.

A triangle with sides a, b, and c.

Formulation

In , Heron's formula, named after , gives the of a when the length of all three sides are known. Unlike other formulas, there is no need to calculate other distances in the triangle first.

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

$A = \sqrt{s(s-a)(s-b)(s-c)},$

where s is the of the triangle; that is,

$s=\frac{a+b+c}{2}.$

Heron's formula can also be written as

$A=\frac{1}{4}\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}$

$A=\frac{1}{4}\sqrt{2(a^2 b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4)}$

$A=\frac{1}{4}\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}$

$A={\frac {1}{4}}{\sqrt {4(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{2}+b^{2}+c^{2})^{2}}}.$

History

The formula is credited to , and a proof can be found in his book, Metrica, written c. CE 60. It has been suggested that 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

$A=\frac1{2}\sqrt{a^2c^2-\left(\frac{a^2+c^2-b^2}{2}\right)^2}$ , where a ≥ b ≥ c,

was discovered by the Chinese independently[] of the Greeks. It was published in Shushu Jiuzhang (“”), written by and published in 1247.

Generalizations

Heron's formula is a special case of for the area of a . Heron's formula and Brahmagupta's formula are both special cases of for the area of a . 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 for the area of a trapezoid or trapezium based only on its sides. Heron's formula is obtained by setting the smaller parallel side to zero.

Expressing Heron's formula with a in terms of the squares of the between the three given vertices,

$A = \frac{1}{4} \sqrt{- \begin{vmatrix} 0 & a^2 & b^2 & 1 \\ a^2 & 0 & c^2 & 1 \\ b^2 & c^2 & 0 & 1 \\ 1 & 1 & 1 & 0 \end{vmatrix} }$

illustrates its similarity to for the of a .

Another generalization of Heron's formula to pentagons and hexagons inscribed in a circle was discovered by .

reference article：

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