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A dihedral angle is defined as the angle between two intersecting planes.

Dihedral angles are tricky to understand and to compute, because we are familiar with the notion of finding the angle between two intersecting lines, not planes.

This problem is solved by turning the angle into an angle between lines. One way is to find the angle between the two normal vectors: this angle must be the same as the dihedral angle. Another way is to take a plane perpendicular to both planes, cut both of the planes, and find the angle between the lines of intersection.

In coordinate geometry, the first method is generally used, since there is a simple description of a plane in terms of its normal vector in coordinates. In synthetic geometry, the second method would usually be used, with a convenient choice of cutting plane.

For example, to find the angle between two faces $ABC$ and $ABD$ of a regular tetrahedron of side length $s$, one would take the cutting plane as the one that contains edge $CD$ and the midpoint $M$ of $AB$. Let $G$ be the centroid of $ABC$. Then, $MD=s\frac{\sqrt{3}}{2}$, and $MG=s\frac{\sqrt{3}}{6}$. By the symmetries of the tetrahedron, $DG$ is perpendicular to the plane containing $ABC$. Thus, the cosine of the dihedral angle equals $\frac{MG}{MD}=\frac{1}{3}$. Thus, the dihedral angle equals $\arccos(\frac{1}{3})$.