the plane-spacing equation
The plane-spacing equation is the formula machine that turns a set of Miller indices and the shape of the unit cell into an actual distance, the d-spacing. You feed in the cell's dimensions and the plane's indices (hkl), and out comes the perpendicular gap between neighbouring planes. There is one version of the formula for each crystal system, because the shape of the cell changes how the geometry works out.
The versions grow more complicated as the cell loses symmetry. For a cubic cell (all edges a, all angles 90 degrees): 1/d^2 = (h^2 + k^2 + l^2) / a^2. For a tetragonal cell (edges a, a, c): 1/d^2 = (h^2 + k^2)/a^2 + l^2/c^2. For an orthorhombic cell (edges a, b, c, all angles 90 degrees): 1/d^2 = h^2/a^2 + k^2/b^2 + l^2/c^2. For a hexagonal cell: 1/d^2 = (4/3)(h^2 + hk + k^2)/a^2 + l^2/c^2. The most general triclinic case, with three different edges and three non-right angles, is a long expression involving all six parameters.
Every one of these can be derived cleanly from the reciprocal lattice, where 1/d is just the length of the reciprocal-lattice vector for (hkl). That is why a powder pattern's peak positions, which give the d-spacings, can be inverted to solve for the unit-cell parameters — the process called indexing. The equation is the bridge from measured angles back to cell geometry.
For a tetragonal crystal with a = 4 angstrom and c = 6 angstrom, the (101) spacing is 1/d^2 = 1/16 + 1/36 = 0.0625 + 0.0278 = 0.0903, so d = 3.33 angstrom. The cubic formula would have given the wrong answer here, because c is not equal to a — the crystal system dictates which version you must use.
Each crystal system has its own spacing formula; the simple cubic one is only a special case.
The tidy cubic form 1/d^2 = (h^2+k^2+l^2)/a^2 is only the special case for equal, orthogonal axes. Using it on a tetragonal, hexagonal or lower-symmetry crystal gives wrong spacings.