“Heron's formula”的意思、由来-开放百科全书

In geometry, Heron's formula (sometimes called Hero's formula), named after Hero of Alexandria,^[1] gives the area of a triangle when the length of all three sides are known. Unlike other formulas, there is no need to calculate other distances in the triangle first.

Formulation

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

Example

Let {{math|△ABC}} be the triangle with sides {{math|a {{=}} 4}}, {{math|b {{=}} 13}} and {{math|c {{=}} 15}}.

The semiperimeter is {{math|s {{=}} {{sfrac|1|2}}(a + b + c) {{=}} {{sfrac|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.

History

The formula is credited to Heron (or Hero) of Alexandria, and a proof can be found in his book, Metrica, written {{circa}} CE 60. It has been suggested that Archimedes knew the formula over two centuries earlier,^[3] 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.^[4]

was discovered by the Chinese independently{{citation needed|date=October 2017}} of the Greeks. It was published in Shushu Jiuzhang (“Mathematical Treatise in Nine Sections”), written by Qin Jiushao and published in 1247.

Proofs

Heron's original proof made use of cyclic quadrilaterals, while other arguments appeal to trigonometry as below, or to the incenter and one excircle of the triangle .

Trigonometric proof using the law of cosines

A modern proof, which uses algebra and is quite unlike the one provided by Heron (in his book Metrica), follows.^[5]

Let {{math|a}}, {{math|b}}, {{math|c}} be the sides of the triangle and {{math|α}}, {{math|β}}, {{math|γ}} the angles opposite those sides.

The altitude of the triangle on base {{math|a}} has length {{math|b sin γ}}, and it follows

Algebraic proof using the Pythagorean theorem

By the Pythagorean theorem we have {{math|b{{sup|2}} {{=}} h{{sup|2}} + d{{sup|2}}}} and {{math|a{{sup|2}} {{=}} h{{sup|2}} + (c − d){{sup|2}}}} according to the figure at the right. Subtracting these yields {{math|a{{sup|2}} − b{{sup|2}} {{=}} c{{sup|2}} − 2cd}}. This equation allows us to express {{math|d}} in terms of the sides of the triangle:

For the height of the triangle we have that {{math|h{{sup|2}} {{=}} b{{sup|2}} − d{{sup|2}}}}. By replacing {{math|d}} with the formula given above and applying the difference of squares identity we get

We now apply this result to the formula that calculates the area of a triangle from its height:

Trigonometric proof using the law of cotangents

From the first part of the Law of cotangents proof,^[7] we have that the triangle's area is both

and {{math|A {{=}} rs}}, but, since the sum of the half-angles is {{sfrac|{{pi}}|2}}, the triple cotangent identity applies, so the first of these is

Numerical stability

Heron's formula as given above is numerically unstable for triangles with a very small angle when using floating point arithmetic. A stable alternative^[8]^[9] involves arranging the lengths of the sides so that {{math|a ≥ b ≥ c}} and computing

The brackets in the above formula are required in order to prevent numerical instability in the evaluation.

Other area formulae resembling Heron's formula

Three other area formulae have the same structure as Heron's formula but are expressed in terms of different variables. First, denoting the medians from sides {{math|a}}, {{math|b}}, and {{math|c}} respectively as {{math|m_a}}, {{math|m_b}}, and {{math|m_c}} and their semi-sum {{math|{{sfrac|1|2}}(m_a + m_b + m_c)}} as {{math|σ}}, we have^[10]

Next, denoting the altitudes from sides {{math|a}}, {{math|b}}, and {{math|c}} respectively as {{math|h_a}}, {{math|h_b}}, and {{math|h_c}}, and denoting the semi-sum of the reciprocals of the altitudes as {{math|H {{=}} {{sfrac|1|2}}(h{{su|b=a|p=−1}} + h{{su|b=b|p=−1}} + h{{su|b=c|p=−1}})}} we have^[11]

Finally, denoting the semi-sum of the angles' sines as {{math|S {{=}} {{sfrac|1|2}}(sin α + sin β + sin γ)}}, we have^[12]

where {{math|D}} is the diameter of the circumcircle: {{math|D {{=}} {{sfrac|a|sin α}} {{=}} {{sfrac|b|sin β}} {{=}} {{sfrac|c|sin γ}}}}.

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 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 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.

Another generalization of Heron's formula to pentagons and hexagons inscribed in a circle was discovered by David P. Robbins.^[13]

Heron-type formula for the volume of a tetrahedron

If {{math|U}}, {{math|V}}, {{math|W}}, {{math|u}}, {{math|v}}, {{math|w}} are lengths of edges of the tetrahedron (first three form a triangle; {{math|u}} opposite to {{math|U}} and so on), then^[14]

See also

References

1. ^{{cite web|title=Fórmula de Herón para calcular el área de cualquier triángulo|url=http://recursostic.educacion.es/descartes/web/materiales_didacticos/formula_heron/formula_de_Heron.htm|language=Spanish|accessdate=30 June 2012}}
2. ^{{cite journal|author=Kendig, Keith|title=Is a 2000-Year-Old Formula Still Keeping Some Secrets?|journal=Amer. Math. Monthly|volume=107|year=2000|pages=402–415|url=http://www.maa.org/programs/maa-awards/writing-awards/is-a-2000-year-old-formula-still-keeping-some-secrets|doi=10.2307/2695295}}
3. ^{{cite book| author=Heath, Thomas L.| title=A History of Greek Mathematics (Vol II)| publisher=Oxford University Press| year=1921| pages=321–323}}
4. ^{{MathWorld |urlname=HeronsFormula |title=Heron's Formula}}
5. ^{{cite book| author=Niven, Ivan| title=Maxima and Minima Without Calculus| publisher=The Mathematical Association of America| year=1981| pages=7–8}}
6. ^{{Cite journal | last = Raifaizen | first = Claude H. | title = A Simpler Proof of Heron's Formula | journal = Mathematics Magazine | volume = 44 | number = 1 | pages = 27–28 | year = 1971}}
7. ^The second part of the Law of cotangents proof depends on Heron's formula itself, but this article depends only on the first part.
8. ^{{cite book |author-first=Pat H. |author-last=Sterbenz |title=Floating-Point Computation |date=1974-05-01 |edition=1st |series=Prentice-Hall Series in Automatic Computation |publisher=Prentice Hall |location=Englewood Cliffs, New Jersey, USA |isbn=0-13-322495-3}}
9. ^{{cite web |url=http://www.cs.berkeley.edu/~wkahan/Triangle.pdf |title=Miscalculating Area and Angles of a Needle-like Triangle |author=William M. Kahan |date=24 March 2000}}
10. ^Benyi, Arpad, "A Heron-type formula for the triangle," Mathematical Gazette" 87, July 2003, 324–326.
11. ^Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle," Mathematical Gazette 89, November 2005, 494.
12. ^Mitchell, Douglas W., "A Heron-type area formula in terms of sines," Mathematical Gazette 93, March 2009, 108–109.
13. ^D. P. Robbins, "Areas of Polygons Inscribed in a Circle", Discr. Comput. Geom. 12, 223-236, 1994.
14. ^W. Kahan, "What has the Volume of a Tetrahedron to do with Computer Programming Languages?", , pp. 16–17.