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Law of cosines generalised for any polygon and any polyhedron

Published online by Cambridge University Press:  23 January 2015

Lee Collins  and
Thomas J. Osler
Lee Collins
Affiliation:
Mathematics Department, Rowan University, Glassboro, NJ 08028 e-mail:osler@rowan.edu
Thomas J. Osler
Affiliation:
Mathematics Department, Rowan University, Glassboro, NJ 08028 e-mail:osler@rowan.edu
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Extract

The law of cosines for any triangle with sides of length a, b and c is c2 = a2 + b2 − 2ab cos γ, where γ is the angle opposite side c. We show that this law generalises nicely to any polygon as well as any polyhedron.

The generalisation to a quadrilateral with sides a, b, c and d is

and for a five sided polygon

Type
Articles
Information
The Mathematical Gazette , Volume 95 , Issue 533 , July 2011 , pp. 240 - 243
Copyright
Copyright © The Mathematical Association 2012

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References

1. Kheyfits, Alexander, The theorem of cosines for pyramids, The College Mathematics Journal, 35 (November 2004) pp. 385–388.Google Scholar