The physics
A point charge sends q/ε0 of flux out, evenly in every direction. A surface gets the share of directions it covers.
A solid angle measures that share. Draw a ball of radius r around the point. A patch of area S on it has Ω = S/r². The whole ball is 4π sr.
share = Ω / 4π cone: Ω = 2π(1 − cos θ)
For a charge on a cube, complete the closed shape. A charge on a face sits in 2 cubes. On an edge it sits in 4, at a corner in 8. Equal pieces get equal shares.
A long wire's lines fan out flat. A strip beside it gets the angle it covers over 360°.