Gear ratio calculator

The formula

v = rpm · π · D / (g · f)

g is the gear's ratio and f the final drive, the fixed reduction in the axle or transaxle; their product is the overall ratio. D is the tyre's overall diameter, so π · D is the distance covered in one turn of the wheel. Engine rpm divided by g · f is wheel rpm, and wheel rpm times the circumference is speed in distance per minute.

In the units people use, that becomes mph = rpm × D(in) ÷ (g × f × 336.14), where 336.14 is the 63,360 inches in a mile divided by 60 minutes and by π. In metric, km/h = rpm × D(m) × π × 60 ÷ 1,000 ÷ (g × f). Solve the same line for rpm and it gives the engine speed at any road speed, which is the cruise column in the calculator.

A metric tyre size carries its own diameter. In 225/45R17, 225 is the section width in millimetres, 45 is the sidewall height as a percentage of that width, and 17 is the rim in inches. The diameter is the rim plus two sidewalls: 17 × 25.4 + 2 × 225 × 0.45 = 634.3 mm, or 24.97 in. That is the unloaded size; a tyre carrying a car squats a few per cent, so real speeds run a little under the calculated ones.

A worked example

Take a six-speed with ratios 3.64, 2.08, 1.36, 1.02, 0.83, 0.69, a 4.105 final drive, 225/45R17 tyres and a 7,000 rpm redline: the gear set of the simulator's compact class, on a common tyre. First gear overall is 3.64 × 4.105 = 14.94, so at 7,000 rpm the wheels turn 468 times a minute and the car does 56 km/h, 35 mph.

Sixth reads 184 mph at redline, which no small hatchback reaches. That figure is geometric. The top two gears are overdrives, there to hold the engine at 2,288 rpm at 60 mph rather than 3,382 in fourth; flat out, air drag stops the car long before the engine reaches redline in them.

What an upshift does to rpm

Road speed does not change across a shift, so the engine drops by the ratio of the two gears: rpm after = rpm before × next gear ÷ current gear. From 7,000 in first (3.64) the shift to second (2.08) lands at 4,000 rpm; the shift from fifth to sixth lands at 5,819. The steps close up toward the top because the low gears are spread to get the car moving and the high ones are packed to keep the engine near its power peak once it is.

That drop is what decides how many gears a vehicle needs. An engine with a narrow power band, like a diesel that stops pulling a few thousand rpm above idle, needs small steps so each shift lands back inside it, which is why a loaded truck carries ten speeds or more. An engine that pulls from low rpm to a high redline gets away with fewer, wider ones.

Gear teeth and torque

For a single pair of gears the ratio is the driven gear's teeth over the driving gear's teeth. A 15-tooth pinion driving a 40-tooth wheel is 2.667:1: 3,000 rpm in becomes 1,125 rpm out, and 200 N·m in becomes 533 N·m out, less a little friction. Power, which is torque times speed, stays the same; a gear trades one for the other.

Put two pairs in series, with the second pair's driving gear on the same shaft as the first pair's driven gear, and the ratios multiply: 15:40 followed by 12:36 is 8:1. That is a compound train, and it is how a gearbox gets a large reduction without a huge gear. An idler, a gear between the driving and driven gears, changes nothing but the direction of rotation, which is how a manual box makes reverse. The calculator's second mode does all three.

Every vehicle class in the simulator

The simulator carries eight vehicle classes, each with a manual gear set, and an automatic for six of them. These are the gear sets it actually runs, with each one's top-gear rpm at cruising speed and the speed every gear reaches at the redline of the engine the class opens with. Speeds at redline are geometric, as above.

The open-wheeler class turns the most in top gear at 60 mph, 5,637 rpm, because its engine revs to 19,250 and the gearing is matched to that. The lowest is the semi Allison 4000 6-speed automatic, at 1,410 rpm. The semi's 10 ratios sit close together because its diesel stops at 2,100 rpm: each shift has to land back inside a narrow band.

Each class link opens that engine in the simulator with that gearbox, where the drag, the clutch and the converter decide what the car really does. How the gears inside a manual, an automatic and a CVT work is a separate page, with each gearbox in 3D.

Questions people ask

Sources

• Thomas D. Gillespie, Fundamentals of Vehicle Dynamics, SAE, chapter 2. Tractive effort through the gearbox and final drive, and why top speed is set by road load, not gearing alone.
• Bosch Automotive Handbook, drivetrain. Overall ratio, gear spacing and the speed-per-rpm relation used above.
• ETRTO and Tire and Rim Association standards for metric tyre designations: section width in millimetres, aspect ratio as a percentage, rim diameter in inches.
• NIST, on the 1959 international yard and pound agreement The inch as exactly 25.4 mm, which fixes the tyre-diameter arithmetic and the mph constant.

Related reading

• Gearbox simulator What is inside a manual, an automatic and a CVT, and how each one changes ratio.
• How a torque converter works The automatic's coupling, and the torque it multiplies before the gears do.
• Horsepower vs torque Why gearing turns one into the other.
• What is engine braking The same ratios, driven from the wheels.

Other calculators

• Compression ratio calculator Static, dynamic and boosted CR from bore, stroke, chamber, gasket and piston.
• Engine displacement calculator Bore × stroke × cylinders to cc, litres and cubic inches, with overbore, and a cc to cubic inch converter.
• Bore and stroke calculator Solve bore or stroke for a target size, the ratio, and piston speed at redline.
• Horsepower calculator Torque and rpm to hp, kW and PS, power from a quarter-mile run, wheel to crank, or an estimate from displacement.
• Dyno correction calculator SAE, STD, DIN and EC corrected power from one reading.

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