Folke Eriksson: On the measure of solid angles, Mathematics Magazine 63,3 (June 1990) 184-187. http://www.maa.org/sites/default/files/Eriksson14108673.pdf Eriksson's formula can be re-written as follows. Let the coordinates of the vertices of the triangle (drawn on unit sphere) be the rows of 3x3 matrix X. Then NonEuclideanSignedArea = 2 * arctan( det(X) / (1+M12+M23+M13) ) where M=X Xtranspose and for spherical geometry K=+1. This formula also works to give the signed area of a hyperbolic nonEuclidean triangle where the first two coordinates of each unit-length 3-vector now are imaginary, and we change arctan to argtanh. And of course in 1 dimension lower (X now a 2x2 matrix) the corresponding formula is Measure = arccos( M12 ) or for hyperbolic use argcosh. But I do not believe there is any formula anywhere near as simple as these if we go to one dimension higher. Specifically mere arctrig will not suffice. Dilogarithms also are needed. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)