Gear Ratio
Ratio, output speed, and output torque from tooth counts.
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The engineering
Gears trade speed for torque at a fixed exchange rate set by tooth count — teeth are the integers that make the kinematics exact, which is why ratio is counted, not measured. Speed divides by the ratio, torque multiplies by it, and power (minus mesh losses) passes through unchanged.
The efficiency box is the number that bites in stacked stages: a single spur mesh is ~98%, but four stages of it is 0.98⁴ ≈ 92%, and a worm drive can burn 30–50% as heat. The torque at the output shaft is what sizes the shaft and keys — multiply before you pick the steel.
Where this math comes from
The Antikythera mechanism — a Greek geared astronomical computer from around 100 BCE — proves toothed wheels and deliberate ratios are ancient engineering, not industrial-age inventions. What antiquity lacked was a tooth shape that transmitted motion smoothly; that took Leonhard Euler, whose 1754 analysis of the involute profile gave gears the curve almost every modern tooth still wears.
Robert Willis's 1841 'Principles of Mechanism' turned gear trains into systematic kinematics — ratios as arithmetic on tooth counts — which is precisely the arithmetic on this card. Everything since has been metallurgy, precision, and lubrication.
- 100 BCEGreek artisans (circa)Antikythera mechanism — geared ratios compute the heavens.
- 1754Leonhard EulerInvolute tooth profile analyzed — the modern gear tooth.
- 1841Robert Willis'Principles of Mechanism' systematizes gear-train kinematics.
See the full timeline of the math behind every calculator →
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