Gear Calculation Formula: RPM, Speed & Torque

This gear calculator works out the relationship between gear ratio, input speed (RPM), and output speed, plus the resulting output torque for a simple two-gear pair. You enter the number of teeth on the driving and driven gears and the input RPM, and you get the gear ratio, output RPM, and torque multiplication. The core formulas are ratio = driven teeth / driving teeth, output RPM = input RPM / ratio, and output torque = input torque × ratio.

Formula

The fundamental gear equation relates the number of teeth on two meshing gears to their rotational speeds:

Gear Ratio (GR) = T_driven / T_driving

Output RPM = Input RPM / GR

Output Torque = Input Torque × GR (assuming 100% efficiency)

Where:

• T_driving = number of teeth on the input (driving) gear
• T_driven = number of teeth on the output (driven) gear
• GR = gear ratio (dimensionless)
• Input RPM = rotational speed of the driving gear (revolutions per minute)
• Output RPM = rotational speed of the driven gear (revolutions per minute)
• Input Torque = torque applied to the driving gear (N·m or lb-ft)
• Output Torque = torque available at the driven gear (same unit as input torque)

For a gear pair, the product of teeth and speed is constant: T_driving × RPM_driving = T_driven × RPM_driven. This is the basis of all gear calculations.

Worked Example

Using the default values from the calculator: driving gear has 20 teeth, driven gear has 40 teeth, input speed is 1000 RPM.

1. Gear ratio = 40 / 20 = 2.00
2. Output speed = 1000 / 2 = 500 RPM
3. Torque multiplier = 2 (so if input torque were 10 N·m, output torque would be 20 N·m)

The calculator shows a gear ratio of 2.00, output speed of 500.0 RPM, and torque multiplier of 2.00. This matches the hand calculation.

When to Use This Formula

Use the gear ratio formula whenever you need to predict the speed or torque change between two meshing gears. Common applications include:

• Bicycle gear ratios (chainrings and cassette sprockets)
• Vehicle transmissions and differentials
• Industrial gearboxes and speed reducers
• Robotics and hobby projects with geared motors
• 3D printer and CNC machine lead screws

The same formula applies to belt and pulley systems if you substitute pulley diameters for tooth counts, and to chain drives if you count sprocket teeth.

How to Read the Results

A gear ratio greater than 1 (e.g., 2.00) means the driven gear turns slower than the driving gear, but with increased torque. This is called a reduction gear. A ratio less than 1 (e.g., 0.5) means the driven gear turns faster but with less torque – an overdrive. A ratio of exactly 1 gives the same speed and torque (a direct drive).

The torque multiplier equals the gear ratio only when you ignore friction. In reality, gear meshing losses (typically 1-2% per gear pair) reduce the output torque slightly. For a single pair, the efficiency is usually 97-99%, so the actual output torque is roughly 0.97 to 0.99 times the ideal value.

Common Mistakes and Misconceptions

• Mixing up which gear is driving: Always identify the input gear first. If you swap the two, you get the inverse ratio.
• Using diameter instead of teeth: For gears, tooth count is proportional to pitch diameter, so you can use either as long as you are consistent. But mixing teeth on one gear and diameter on the other gives wrong results.
• Forgetting that torque and speed are inversely related: You cannot increase both at the same time. Power (torque × angular speed) stays roughly constant (minus losses).
• Ignoring units: RPM is the same in any system, but torque must be in consistent units (N·m with N·m, lb-ft with lb-ft).

Limits of the Formula

The simple gear ratio formula assumes ideal geometry and ignores friction, gear tooth deflection, and backlash. It also assumes the gears are rigid and the shafts are perfectly aligned. For high-precision applications, you would need to account for:

• Efficiency losses (typically 1-2% per gear pair)
• Inertia of the gears and attached masses
• Tooth profile and contact ratio
• Lubrication and operating temperature

The formula gives the steady-state speed and torque, not the transient behavior during acceleration. For dynamic analysis, you need the moment of inertia and the torque-speed curve of the motor.

Frequently Asked Questions

How do I calculate gear ratio from RPM?

If you know the input and output RPM, the gear ratio is simply input RPM divided by output RPM. For example, 3000 RPM in and 1000 RPM out gives a ratio of 3. This is useful when you do not know the tooth counts.

What is the difference between gear ratio and gear reduction?

Gear reduction specifically refers to a ratio greater than 1, which reduces speed and increases torque. Gear ratio is a general term that includes overdrive (ratio less than 1) and direct drive (ratio = 1).

Does gear ratio affect horsepower?

No, the gear ratio does not change the power. Power is torque times angular speed. If torque doubles and speed halves, the product remains the same (ignoring losses). Gears trade speed for torque, but the power stays constant.

How do I calculate the output torque of a gearbox?

Multiply the input torque by the gear ratio, then multiply by the efficiency (usually 0.95-0.99 per gear pair). For example, 10 N·m input, ratio 3, efficiency 0.97 gives 10 × 3 × 0.97 = 29.1 N·m output.

Can I use the same formula for belt and pulley systems?

Yes, replace the number of teeth with the pitch diameter of the pulleys. The ratio is driven pulley diameter divided by driving pulley diameter. The rest of the calculations are identical.

What is the center distance between two gears?

The center distance is the sum of the pitch radii of the two gears. For standard gears, it is (T1 + T2) × module / 2, where module is the gear size in millimeters. For imperial gears, use pitch diameter = teeth / diametral pitch.

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in t1 | Driving gear teeth | teeth | 20 | 1
in t2 | Driven gear teeth | teeth | 40 | 1
in rpm | Input speed | RPM | 1000 | 0
out ratio | Gear ratio | - | t2 / t1 | 2
out out_rpm | Output speed | RPM | rpm / ratio | 1
out torque_mult | Torque multiplier | - | ratio | 2