The difference between the sum of the angles spherical triangle
and
The notation surface area of a spherical
triangle (Zwillinger 1995, p. 469). The notation 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 l'Huilier's theorem ,
where semiperimeter .
See also 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. Harris, J. W. and Stocker, H. Handbook
of Mathematics and Computational Science. 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. Referenced on Wolfram|Alpha Spherical Excess
Cite this as:
Weisstein, Eric W. "Spherical Excess."
From MathWorld https://mathworld.wolfram.com/SphericalExcess.html
Subject classifications
, 
, and 
of a spherical triangle
and 
radians (
),
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).
, 
, and 
is known as l'Huilier's theorem,
![tan(1/4E)=sqrt(tan(1/2s)tan[1/2(s-a)]tan[1/2(s-b)]tan[1/2(s-c)]),](/images/equations/SphericalExcess/NumberedEquation2.svg)
is the semiperimeter.
