Law of Cosines

Pronunciation: /lɔ ʌv ˈkoʊ.saɪnz/ Explain

The law of cosines relates the angles of a triangle to the length of the sides: c2 = a2 + b2 - 2ab cosγ.. The law of cosines is a generalization of the pythagorean theorem. Since cos(90°) is zero, the formula is c2 = a2 + b2 for right triangles.

A geometric equivalent of the law of cosines was described in Euclid's Elements, Book 2, proposition 12 and proposition 13. The law of cosines can also be used to generate the formula for the angle opposite side c, if one knows the lengths of the sides:

gamma = arccosine ( a squared plus b squared minus c squared) / (2ab).

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Manipulative 1 - Law of Cosines Created with GeoGebra.


  1. McAdams, David E.. All Math Words Dictionary, law of cosines. 2nd Classroom edition 20150108-4799968. pg 106. Life is a Story Problem LLC. January 8, 2015. Buy the book

More Information

  • McAdams, David E.. Triangle. All Math Words Encyclopedia. Life is a Story Problem LLC. 8/31/2018.
  • Euclid of Alexandria. Elements. Clark University. 9/6/2018.

Cite this article as:

McAdams, David E. Law of Cosines. 4/23/2019. All Math Words Encyclopedia. Life is a Story Problem LLC.

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Revision History

4/23/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
8/31/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
7/18/2018: Initial version. (McAdams, David E.)

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