Angle between two planes ($\theta =\arccos \frac{|{\mathbf{n}}_1\cdot {\mathbf{n}}_2|}{|{\mathbf{n}}_1||{\mathbf{n}}_2|}$)

The acute angle between two unoriented planes is the acute angle between their nonzero normal vectors, giving the displayed formula. For an oriented convex polyhedron, the interior dihedral angle of a convex polyhedron is instead $\pi$ minus the angle between outward face normals. The distinction matters when a face normal is reversed: its unoriented plane is unchanged, but its vector angle is replaced by its supplement.

For smooth level surfaces $f=0$, $g=0$ meeting at a point where both gradients are nonzero, their acute angle is the angle between two planes tangent to the surfaces there. Each gradient is a normal vector to its tangent plane, proving the displayed formula. Vanishing gradients require further analysis because the level-set equation then need not specify a unique tangent plane.