the interplanar angle
Two crystal planes tilted with respect to each other make an angle, and often you need to know it — to identify a facet, to interpret an etch pit, or to plot poles on a stereographic projection. The angle between two planes is defined as the angle between their normals (the arrows sticking straight out of each), which is the same as the angle you would measure between the planes themselves.
For a cubic crystal there is a clean dot-product formula. If the planes are (h1 k1 l1) and (h2 k2 l2), then cos phi = (h1 h2 + k1 k2 + l1 l2) / (sqrt(h1^2 + k1^2 + l1^2) times sqrt(h2^2 + k2^2 + l2^2)). For example the angle between (100) and (110): the top is 1, the bottom is 1 times sqrt(2), so cos phi = 0.707 and phi = 45 degrees. The angle between (111) and (100): cos phi = 1 / sqrt(3) = 0.577, giving the famous 54.7 degrees that shows up all over cubic crystallography.
The convenience of this formula is a cubic-only luxury. In cubic crystals the plane normal points along the like-indexed direction, so the plane angle equals the direction angle and both use the same simple sum. In any lower-symmetry system the normal is generally not along [hkl], and you must use the reciprocal-lattice metric instead of the naive dot product — a real trap if you forget which system you are in.
In a cubic crystal the angle between (111) and (100) is arccos(1/sqrt(3)) = 54.7 degrees, and between (100) and (110) it is arccos(1/sqrt(2)) = 45 degrees. These fixed angles are why the facets of a cubic crystal always meet at the same recognisable tilts.
The cubic dot-product gives the angle between planes via their indices — but only for cubic crystals.
The simple dot-product formula is valid only in cubic crystals, where the plane normal lies along [hkl]. For other systems use the reciprocal-lattice metric, or you will get wrong angles.