the dihedral angle

Open a book and the two covers meet along the spine, making an angle between them; fold a sheet of paper and the crease has a sharpness you could measure. That opening between two flat half-planes meeting along a common edge is a dihedral angle. It is the three-dimensional cousin of the ordinary angle between two lines, and it is the natural way to measure how two planes are tilted relative to each other.

Two planes that are not parallel meet in a line. The dihedral angle is measured in a slice taken perpendicular to that line of intersection: in that cross-section each plane shows up as a ray from the edge, and the angle between those rays is the dihedral angle. The fast way to compute it uses normal vectors. If the planes have normals n_1 and n_2, then cos theta = (n_1 . n_2) / (|n_1| |n_2|). Because each plane has two opposite normals, this can return the angle or its supplement, so one usually takes the acute version by using absolute value: cos theta = |n_1 . n_2| / (|n_1| |n_2|). Special cases fall out at once: the planes are parallel when the normals are parallel (theta = 0), and perpendicular when the normals are perpendicular, i.e. n_1 . n_2 = 0.

Dihedral angles are everywhere geometry meets the solid world — the pitch of a roof, the angle of a folded box flap, the wedge angle in crystals and molecules. The one trap to keep in mind is the two-normals ambiguity: the angle between the normals might be the angle you want OR its supplement (they sum to 180 degrees). Decide up front whether you want the acute opening or the obtuse one, and use the absolute value or a sketch to choose correctly. The dot-product formula gives a number; you supply the geometric judgement about which opening is meant.