When a physical relation involves dimensional variables, dimensional analysis can reduce it to a relation among fewer dimensionless groups. The Buckingham Pi theorem states that if a dimensionally homogeneous problem contains $n$ dimensional variables built from $r$ independent fundamental dimensions, then the relation can be expressed using $n-r$ independent dimensionless combinations.

Suppose the drag force $F$ on a sphere depends on fluid density $\rho$, speed $v$, diameter $D$ and viscosity $\mu$: $$F=f(\rho,v,D,\mu).$$ There are five variables and three fundamental dimensions $M,L,T$, so two independent dimensionless groups are expected. One convenient choice is $$\Pi_1=\frac{F}{\rho v^2D^2},$$ which is a drag coefficient up to conventional constants, and $$\Pi_2=\frac{\rho vD}{\mu}=\mathrm{Re},$$ the Reynolds number. The unknown dimensional relation must therefore have the form $$\frac{F}{\rho v^2D^2}=\Phi(\mathrm{Re}).$$

Instead of experimentally exploring four independent dimensional inputs, one can study how one dimensionless response varies with one dimensionless control parameter.

The theorem does not determine the function $\Phi$ or reveal all relevant physics; missing variables lead to incomplete groups. Its power is structural: it identifies the minimum dimensionless parameter space compatible with dimensional consistency and underlies similarity laws in fluids, heat transfer, mechanics and experiments.