Math 2.0

Geometry Triangles and circles Theorem

Heron's formula

Builds on Area of a triangleTriangle

Statement

Let ABCABC be a triangle in the Euclidean plane, with the side lengths aa, bb and cc. Let s=12(a+b+c)s = \frac{1}{2}(a + b + c) be its semiperimeter. Then the area TT of the triangle is T=s(s−a)(s−b)(s−c).T = \sqrt{s (s - a)(s - b)(s - c)}.

Remarks

An equivalent form without ss is 16T2=(a+b+c)(−a+b+c)(a−b+c)(a+b−c)16 T^2 = (a + b + c)(-a + b + c)(a - b + c)(a + b - c). For example, the triangle with the sides 44, 1313 and 1515 has s=16s = 16 and T=16⋅12⋅3⋅1=24T = \sqrt{16 \cdot 12 \cdot 3 \cdot 1} = 24. One proof uses the law of cosines to get cos⁡γ\cos \gamma from the sides, and then T=12absin⁡γT = \frac{1}{2} a b \sin \gamma.

History

The formula carries the name of Heron of Alexandria, who gave a proof in his Metrica in the 1st century. It was possibly known earlier. Qin Jiushao published an equivalent formula in 1247.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Triangle, concept, no evidence levelTriangleconceptArea of a triangle, concept, no evidence levelArea of a triangleconceptHeron's formula, statement, no evidence levelHeron's formulastatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Triangle concept
  2. Area of a triangle concept
  3. Heron's formula statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Heron%27s_formula, lead section
About this page
Tag0FRBZ2X4HJ
URIm2://0FRBZ2X4HJ
Aliasesheros-formula
Kindstatement, theorem
WritingW1 stated
MSC51M25
StatusThis copy of the site has no build data. Run m2 status 0FRBZ2X4HJ.
UsesArea of a triangle, Triangle
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