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By Girard's Spherical Excess Formula, a spherical triangle on unit sphere with angles $A, B, C$ has area $$ A + B + C - \pi. $$ I would like to know, if there is a generalization for this formula to higher dimensions.

There is a thread on MathOverflow about calculating the volume of elliptical tetrahedron. It seems that for odd dimensions the problem is much complex. But one answer in that thread suggests that it is easier to get one for even dimensions. However I could not find any result through Google.

I really know very little about geometry. Any information regarding this problem would be appreciated.

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See these notes by J. G. Heckman (he focuses on the hyperbolic case, but the spherical case is essentially identical).

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    $\begingroup$ The paper gives Reduction Formula for computing the volume of simplex of even dimension. But that formula depends on calculating volumes of simplici of all lower dimensions, not just lower even dimensions. Does this mean that this problem is not really solved for even dimensions also? $\endgroup$
    – LeafGlowPath
    Commented May 2, 2014 at 14:23
  • $\begingroup$ It is an exercise to drop out the odd-dimensional terms... $\endgroup$
    – Igor Rivin
    Commented May 3, 2014 at 1:59
  • $\begingroup$ @HassanJolany What is your point, exactly? $\endgroup$
    – Igor Rivin
    Commented May 14, 2014 at 20:03

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