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# Spherical Excess

The difference between the sum of the angles , , and of a spherical triangle and radians (),

The notation is sometimes used for spherical excess instead of , which can cause confusion since it is also frequently used to denote the surface area of a spherical triangle (Zwillinger 1995, p. 469). The notation is also used (Gellert et al. 1989, p. 263).

The value of the excess is the solid angle (in steradians) subtended by the spherical triangle, as proved by Thomas Hariot in 1603 (Hopf 1940).

The equation for the spherical excess in terms of the side lengths , , and is known as l'Huilier's theorem,

where is the semiperimeter.

Angular Defect, Descartes Total Angular Defect, Girard's Spherical Excess Formula, L'Huilier's Theorem, Spherical Triangle, Tetrahedron

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## References

Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). VNR Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold, 1989.Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 109, 1998.Hopf, H. "Selected Chapters of Geometry." ETH Zürich lecture, pp. 1-2, 1940. http://www.math.cornell.edu/~hatcher/Other/hopf-samelson.pdf.Todhunter, I. and Leathem, J. G. "Spherical Trigonometry: For the Use of Colleges and Schools." London: Macmillan, p. 101, 1901.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 469, 1995.

Spherical Excess

## Cite this as:

Weisstein, Eric W. "Spherical Excess." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SphericalExcess.html