MecMovies 4.0 : M6.9: Gear Basics

Let $C$ = gear circumference, $p$ = pitch of a gear tooth (i.e., arclength of one tooth), $N$ = number of teeth on the gear, and $D$ = gear diameter. The circumference of a gear can be expressed either in terms of its number of teeth $N$ or its diameter $D$ .

$C = pN = πD$

Therefore, for the large and small gears:

$C_{large} = {pN}_{large} = {πD}_{large} C_{small} = {pN}_{small} = {πD}_{small}$

Solve for $p$ in each equation and equate:

$p = \frac{{πD}_{large}}{N_{large}} = \frac{{πD}_{small}}{N_{small}}$

$∴ \frac{D_{large}}{D_{small}} = \frac{N_{large}}{N_{small}} ∴ \frac{D_{small}}{D_{large}} = \frac{N_{small}}{N_{large}}$