I CCed some people a faulty formula for the solid angle at the vertices of the base of a regular n-gonal pyramid. It worked for regular tetrahedra, and perhaps nothing else. Here's one (of many) that works for a regular n-gon base: pyramidSolidAngles[n_, 𝜃_] :={2 π - 2 n ArcTan[Cos[𝜃] Tan[π/n]], 2 ArcSec[(1 + Sin[π/n] Sin[𝜃])/(Sin[π/n] + Sin[𝜃])]} (𝜃 is the ∠ between the altitude and the generator.) There are bewilderingly many formulæ for the solid angle at a trihedral vertex with angles a, b, c. Here are five: solidAngle[a_, b_, c_] := {2 ArcTan[Sqrt[1 - Cos[a]^2 - Cos[b]^2 - Cos[c]^2 + 2 Cos[a] Cos[b] Cos[c]]/(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)])]} FullSimplify can't begin to equate them. Question: For a variable apex above a fixed base-polygon, what is the locus of that apex which preserves the solid angle? Would it be spherical, analogous to the 2D case? It's hard to see how it wouldn't be at simplest ellipsoidal, in the case of an extremely elongated base. I just realized something obvious. I've only been thinking about the solid angles at the vertices of polyhedra. But you can define a (4π supplementary) pair of solid angles for any point on a piecewise smooth surface. It assigns a sort of "topological curvature", bounded between 0 and 4π, to many points where the usual curvature blows up. At a point of bounded curvature, both solid angles are 2π. For a finite isosceles cone with angle 𝜃 between its altitude and generators, the interior solid angles at the apex and and rim are (sending n→∞ in the pyramid) coneSolidAngles[𝜃_] := {2 (1 - Cos@𝜃) π, π - 2 𝜃} I.e., the solid angle at the rim has twice the magnitude of the planar angle there. Which is also true of the smooth areas, but not the apex. --rwg Because 4π ~ 12, one face of a dodecahedron subtends only a little more than a steradian. gosper.org/1steradian.png Scandal: Due to insidious infiltration by Egyptians, Mathematica's Pyramid primitive permits only quadrilateral bases.