Two complexes can contain exactly the same atoms and still be different compounds, because the ligands sit in different places around the metal. Using the shapes from coordination compounds, you can count these isomers.
Key idea
Geometric isomers differ in which positions like ligands occupy: cis (side by side, 90° apart) or trans (opposite, 180°); in octahedra with three of a kind, fac (all on one face) or mer (around a meridian). Optical isomers (enantiomers) are mirror images that cannot be superimposed; such a complex is chiral. Rotating a complex never makes a new isomer.
Why it works
Isomers can be turned into each other only by breaking bonds and rebuilding them in new positions. Turning a model in space keeps every ligand next to the same neighbors, so it is the same complex seen from another side. In a square or an octahedron, two positions can be adjacent or opposite, so two like ligands give cis and trans. In a tetrahedron every pair of corners is adjacent, so $\mathrm{MA_2B_2}$ has only one form. A complex with no mirror plane, such as $\mathrm{[Co(en)_3]^{3+}}$, a three-bladed propeller, has a left-handed and a right-handed form.
Worked examples
Worked example: cis and trans
How many geometric isomers does octahedral $\mathrm{[Co(NH_3)_4Cl_2]^+}$ have?
- The two chloride ligands can be adjacent (90°) or opposite (180°).
- Every other placement turns into one of these by rotation.
Answer: Two: cis and trans.
Worked example: fac and mer
How many geometric isomers does octahedral $\mathrm{[Co(NH_3)_3Cl_3]}$ have?
- The three chloride ligands can cover one triangular face (fac: each Cl 90° from the other two).
- Or they can run around a meridian (mer: two of them 180° apart, the third between).
Answer: Two: fac and mer.
Worked example: Counting mirror images
How many stereoisomers does $\mathrm{[Co(en)_2Cl_2]^+}$ have in all?
- Geometric: the chlorides are trans or cis.
- trans has a mirror plane: not chiral, one form. cis has no mirror plane: a pair of mirror images.
- Total: $1 + 2 = 3$ stereoisomers.
Answer: Three.
Common mistake
Counting rotated copies of one complex as different isomers. Check by asking a simple question about each drawing: are the two like ligands adjacent or opposite? If two drawings give the same answers, they are the same isomer.
Geometry can matter a great deal. cis-$\mathrm{[Pt(NH_3)_2Cl_2]}$, cisplatin, is a widely used cancer drug; the trans isomer is not.