RWG: solidAngle[a_, b_, c_] := { 2 ArcTan[Sqrt[ 1 - Cos[a]^2 - Cos[b]^2 + 2 Cos[a] Cos[b] Cos[c] - Cos[c]^2]/( 1 + Cos[a] + Cos[b] + Cos[c])], ArcCos[-1 + (1 + Cos[a] + Cos[b] + Cos[c])^2/((1 + Cos[a]) (1 + Cos[b]) (1 + Cos[c]))], 2 ArcCos[1/ 4 (1 + Cos[a] + Cos[b] + Cos[c]) Sec[a/2] Sec[b/2] Sec[c/2]], 2 ArcSin[1/(4 Sqrt[2]) Sqrt[-1 - Cos[2 a] - Cos[2 b] + 4 Cos[a] Cos[b] Cos[c] - Cos[2 c]] Sec[a/2] Sec[b/2] Sec[c/2]], 4 ArcTan[\[Sqrt](Tan[1/4 (a + b - c)] Tan[1/4 (a - b + c)] Tan[ 1/4 (-a + b + c)] Tan[1/4 (a + b + c)])]} DanA's would make a sixth, but I can't make sense of it. --WDS; Re Asimov, you appear to be confused about lengths versus angles. Given 3 points on unit sphere, the edge lengths a,b,c are the geodesic distances between them; the (dihredral) angles A,B,C are the angles formed on the sphere at the corners of that triangle with geodesics as edges. Of course these are related. Asimov's formula, also called Albert Girard's formula since apparently it first was known to Thomas Harriot, is area=A+B+C-pi. Of the formulas RWG lists, most were also listed in appendix C of my http://rangevoting.org/BestVrange.html#appC where I listed 7 such formulae due to Cagnoli, Lagrange, Euler, Lhuiler, and Todhunter all before 1860. But RWG's ArcCos[ -1 + (1 + Cos[a] + Cos[b] + Cos[c])^2 / ((1 + Cos[a]) (1 + Cos[b]) (1 + Cos[c])) ] was not previously known to me and might be new. It might be interesting to perform a mildly exhaustive computer search for all such formulae. I programmed such a search, but it may be buggy; if it works it will need to run overnight. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)