I'm trying to find the angle between two sides drawn from center for a tetrahedron.
I assumed that each face makes solid angle $ \pi$ at the center and angle between three sides is equal. So I did the following $$\int_0^x\int_0^x \sin \theta \; d\theta \;d\phi = \pi $$ I am assuming that the surface is equal to solid angle in this figure if radius is 1
I am getting $x = 2.094 \approx 120 $, degrees. I should be getting close to 109.
Furthermore I'm trying to generalize it to any solid angle made by three sides having equal angle between them. Can you suggest what assumption did I make wrong? Or taking tetrahedron as particular example was incorrect?