Step 1: Use the steric number (SN) approach. The steric number of the central atom is the number of atoms directly bonded to it plus the number of lone pairs on it: $SN = (\text{bonded atoms}) + (\text{lone pairs})$.
Step 2: In $XeF_6$, xenon is bonded to 6 fluorine atoms. To find the lone pairs on Xe, take Xe's 8 valence electrons and subtract the 6 electrons it puts into the 6 Xe-F bonds: $8 - 6 = 2$ electrons, which is 1 lone pair. So $SN = 6 + 1 = 7$.
Step 3: A steric number of 7 corresponds to a pentagonal bipyramidal arrangement of electron domains around Xe, meaning 5 positions in an equatorial pentagon and 2 axial positions, with the lone pair sitting in one of these 7 slots.
Step 4: Since only 6 of these 7 positions actually hold a fluorine atom (the 7th holds the lone pair), the molecule's atomic shape is not the full pentagonal bipyramid, and it is also not a perfect octahedron, because the lone pair still exerts extra repulsion that pushes the 6 F atoms out of ideal octahedral angles. The net result, confirmed experimentally, is a distorted (irregular) octahedral shape for the 6 fluorine atoms around xenon.
\[\boxed{\text{Distorted octahedral}}\]