(Geometry, especially Solid Geometry:)

"The" angle between two planes or other planar elements in three dimensional euclidian space **R**^{3}, usually faces of a polyhedron which meet along an edge. Unlike a solid angle, only 2 planes are involved in a "dihedral" angle.

Suppose we have 2 planes α and β in **R**^{3}, which meet along a line l=α∩β. What's the angle between them? Angles are inherently two dimensional objects, so we must reduce our problem to measuring some angle along some plane. Any plane γ not parallel to either of α,β will intersect each of them along a straight line. Thus, γ∩α and γ∩β are straight lines on the plane γ, which intersect at the point γ∩l. We can measure ∠(γ∩α,γ∩β) to get *an* angle between α and β.

Unfortunately, a little thought will show that this angle depends on the choice of γ! Which γ should we choose? A good idea in geometry is, other things being equal, to take something orthogonal.

**Definition.** The *dihedral angle* between α and β is the angle ∠(γ∩α,γ∩β), when γ is the plane perpendicular to l.

To give some taste of why this is a good idea, here are some equivalent definitions:

- The minimum possible angle ∠(γ∩α,γ∩β) for all possible choices of γ.
- The angle between the normals to the planes α,β through a point on l=α∩β.
- The angle between the projections P(α) and P(β) when P is the projection onto a plane along l.

With so many equivalent formulations, it has got to be good for something. Since it's not

*clearly* "THE" angle, we qualify by calling it the "dihedral angle".