Gear ratio simulator

Drag the sliders. The gears are real involute gears (20° pressure angle), drawn from the same model as our Shorts, so the teeth mesh exactly as shown.

The formulas

ω₂ = ω₁ × z₁ / z₂   per stage, multiplied across stages

Where two gears touch, their teeth move at the same speed: v = ω₁r₁ = ω₂r₂. The pitch radius is proportional to the number of teeth (r = m·z/2 for module m), so the speeds are in the inverse ratio of the teeth.

T_out = T_in × (z₂ / z₁)^stages   (no friction)

Power is torque × speed, and an ideal gear neither makes nor loses power, so whatever speed you give up comes back as torque. Real spur gears lose a little to friction at each mesh.

Each meshing pair turns in opposite directions, so the output direction flips with every stage. With several stages, the next small gear is fixed to the same axle as the big gear before it, in a different layer.

ε = (√(r_a1² − r_b1²) + √(r_a2² − r_b2²) − a·sin α) / (π m cos α)

The contact ratio ε is the average number of tooth pairs touching. Above 1, a new pair takes over before the last one lets go, so the motion is smooth. Gears with fewer than 17 teeth at 20° need their tooth roots cut back (undercut), so the sliders start at 17.