coordination geometry
Once you know how many groups crowd a metal, the next question is how they arrange themselves in space. They do not pile up randomly — like balloons tied together pushing apart, the ligands spread out to keep as far from one another as possible. The resulting shape is the coordination geometry, and a few standard shapes cover most of chemistry.
The geometry follows mainly from the coordination number. Two-coordinate complexes are usually linear (180 degrees), seen with d10 ions like [Ag(NH3)2]+ and [Au(CN)2]-. Four-coordinate complexes come in two flavors: tetrahedral, the default, with ligands at the corners of a tetrahedron about 109.5 degrees apart (as in [CoCl4]2- or [Ni(CO)4]); and square planar, where the four ligands sit at the corners of a flat square, favored by d8 metals such as Pt(II), Pd(II), Ni(II), and Au(III) because of how their d orbitals fill. Six-coordinate complexes are overwhelmingly octahedral, with ligands at the six vertices of an octahedron (think up, down, and four around the equator) — by far the commonest geometry of the whole field. Five-coordinate gives trigonal bipyramidal or square pyramidal, two shapes close in energy that often interconvert. Higher numbers (7, 8, 9) give more complex polyhedra, mostly with large lanthanide and actinide ions.
Geometry is the stage on which isomerism plays out: an octahedral MA4B2 can be cis or trans, an octahedral MA3B3 can be fac or mer, and a tetrahedral or octahedral complex with the right ligands can be chiral. Geometry also governs reactivity and, through the way the d orbitals split in each shape, the color and magnetism. A subtle point: which four-coordinate shape a metal chooses (tetrahedral vs square planar) is often decided by d-electron count and ligand-field effects, not by size alone.
Nickel(II) with chloride gives tetrahedral [NiCl4]2- (blue), but with cyanide it gives square planar [Ni(CN)4]2- (yellow) — same coordination number 4, two different geometries, because the strong-field cyanide drives the d8 nickel into the square-planar shape.
Same coordination number 4, but ligand-field effects decide tetrahedral versus square planar.
Coordination number does not pin down the shape by itself. Four-coordinate can be tetrahedral or square planar, and five-coordinate can flip between trigonal bipyramidal and square pyramidal — d-electron count and ligand field tip the balance.