How exactly are brushes created from planes?

[Kein Mitleid]

Does anyone here know exactly how Valve goes about turning planes into solids? I know three planes need to intersect to form a vertex, and I know the math behind all this, but I'm visualizing some scenarios, based on my limited knowledge, in which the vertices aren't correctly predicted.

For instance, take a three-dimensional trapezoid. It's flat on the top, but if I'm correct about planes and vertices, the slanted sides and adjacent side of the trapezoid should converge to form an edge at the top, turning the trapezoid into a triangular prism. How come this doesn't happen in Hammer?

[zombie@computer]

Each brushside of a brush starts as a near-infinitely sized plane. Then, we simply iterate all other brushsides and cut away the part of that plane that is 'outside' the brush. What is left is a correctly shaped brush.

[Kein Mitleid]

Thank you so much! Never thought about it like that. I just realized I also need to take into account the direction the face is facing too, so I can define the actual plane area.