Heron's formula

From Wikipedia, the free encyclopedia

A triangle with sides a, b, and c.
A triangle with sides a, b, and c.

In geometry, Heron's formula (also called Hero's formula) states that the area (A) of a triangle whose sides have lengths a, b and c is

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

where s is the semiperimeter of the triangle:

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

Heron's formula can also be written as

A={\ \sqrt{(a+b+c)(a+b-c)(b+c-a)(c+a-b)\,}\ \over 4}.\,

Contents

The formula is credited to Heron of Alexandria, and a proof can be found in his book, Metrica, written c. 60 AD. 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. [1]

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

\cos(C) = \frac{a^2+b^2-c^2}{2ab}

by the law of cosines. From this we get with some algebra

\sin(C) = \sqrt{1-\cos^2(C)} = \frac{\sqrt{4a^2 b^2 -(a^2 +b^2 -c^2)^2 }}{2ab}.

The altitude of the triangle on base a has length bsin(C), and it follows

S\, = \frac{1}{2} (\mbox{base}) (\mbox{altitude})
= \frac{1}{2} ab\sin(C)
= \frac{1}{4}\sqrt{4a^2 b^2 -(a^2 +b^2 -c^2)^2}
= \frac{1}{4}\sqrt{(2a b -(a^2 +b^2 -c^2))(2a b +(a^2 +b^2 -c^2))}
= \frac{1}{4}\sqrt{(c^2 -(a -b)^2)((a +b)^2 -c^2)}
= \frac{1}{4}\sqrt{(c -(a -b))((c +(a -b))((a +b) -c))((a +b) +c)}
= \sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}.

The difference of two squares factorization was used in two different steps.

Heron's formula as given above is numerically unstable for triangles with a very small angle. A stable alternative[2] involves arranging the lengths of the sides so that: a ≥ b ≥ c and computing

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

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

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

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.

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

S = \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 Tartaglia's formula for the volume of a three-simplex.

  1. ^ http://mathworld.wolfram.com/HeronsFormula.html
  2. ^ http://http.cs.berkeley.edu/~wkahan/Triangle.pdf
  • Heath, Thomas L. (1921). A History of Greek Mathematics (Vol II). Oxford University Press, 321-323. 

Retrieved from "http://en.wikipedia.org../../../h/e/r/Heron%27s_formula.html"
Advanced Search
Included Web Search Engines


Safe Search

close

Top Matching Results

Occasionally Search.com will highlight specialized results that are based on the context of your query. Examples of specialized results include specific links to news, images, or video.

Top Matching Results may highlight information from other Search.com pages, content from the CNET Network of sites, or third party content. The listings are based purely on relevance. Search.com does not receive payment for listings in this section but our partners that provide this data may get paid for listing these products.

Sponsored Links

This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.

Search Results

Search.com sends your search query to several search engines at one time and integrates the results into one list which has been sorted by relevance using Search.com's proprietary algorithm. You can customize the list of search engines included in your metasearch from the preferences.

The search engines that are used in your metasearch may allow companies to pay to have their Web sites included within the results. To view the Paid Inclusion policy for a specific search engine, please visit their Web site. Search.com does not accept payment or share revenue with any search engine partner for listings in this section.