Advanced Geometry by H.C. Rajpoot-2014

Using the Law of Cosines and area by Spherical Excess in spherical trigonometry, given that the angles at the vertices are $\dfrac { \pi }{ 3 }$ , the solid angle (which is the area of the spherical triangle as defined by the $3$ planes of the tetrahedron meeting at the center of an unit sphere) is

$3Cos(\dfrac { \pi }{ 3 } )-\pi =0.551286...$