(Geometry, especially Solid Geometry:)
"The" angle between two planes or other planar elements in three dimensional euclidian space R3, 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 R3, 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".

### Examples

• The dihedral angle between faces of a cube is π/2=90°.
• The dihedral angles of a prism formed by a polygon P are the angles of P, and π/2=90° between each side and the top.
• The dihedral angle between 2 adjacent faces of an icosahedron are arccos(-sqrt(5)/3) [MathWorld].

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