RPM Calculator
Convert RPM to angular velocity, surface speed, and frequency, or work out the output speed of a pulley system, gear drive, or driven wheel. Pick a mode, fill in the values you know, and get the result you need with a full worked explanation.
Formula
Worked example
At 3000 RPM with a 0.3 m radius: omega = 2pi x 3000 / 60 = 314.16 rad/s; surface speed v = 314.16 x 0.3 = 94.25 m/s. For a pulley system: a 100 mm driver turning at 1450 RPM drives a 200 mm pulley at 1450 x (100/200) x (1 - 0.02) = 710.5 RPM after 2% slip. For machining: a 25 mm end mill in aluminium at 300 m/min needs N = (300 x 1000) / (pi x 25) = 3820 RPM.
From RPM to angular velocity and frequency
Revolutions per minute (RPM) is the everyday way to describe how fast something spins. Physics and engineering equations, however, prefer two other measures: angular velocity in radians per second (rad/s) and frequency in hertz (Hz, meaning revolutions per second). Because one full turn is 2pi radians and a minute is 60 seconds, the conversion is omega = 2pi x N / 60. Frequency is simply N / 60. Degrees per second, useful for servo control and robotics, follows from multiplying rad/s by 180 / pi. This calculator handles all four representations in one step.
Surface speed and why radius matters
Every point on a rigid spinning body shares the same angular velocity. The distance it actually travels per second, called surface speed or linear velocity, depends on how far the point sits from the axis: v = omega x r. A grinding wheel rim at large radius moves far faster than a point near the hub at identical RPM. That is why manufacturers specify maximum surface speeds for abrasive wheels (for safety) and recommended surface speeds for cutting tools (for tool life). Surface speed also determines tyre-to-road contact patch speed, conveyor belt speed, and the pitch speed of gears.
Pulley and belt drive calculations
A belt drive links two pulleys of potentially different diameters. Because the belt moves at the same speed at both pulleys, the output RPM is input RPM multiplied by the ratio of driver to driven diameter: N2 = N1 x D1 / D2. A smaller driven pulley spins faster; a larger one spins slower. Real V-belts slip by about 1-3% due to elastic deformation, reducing the driven speed slightly. Toothed (synchronous) belts eliminate slip entirely. Belt speed itself matters too: most V-belts are rated to about 25-30 m/s, above which centrifugal forces reduce tension and efficiency.
Gear ratio and output shaft speed
Gears transmit rotation through meshing teeth. Because both gears share the same pitch line velocity, output RPM equals input RPM divided by the gear ratio, where the ratio is the number of driven teeth divided by driver teeth: N_out = N_in / (T_driven / T_driver). A ratio greater than one is a speed reduction and a torque multiplication; less than one is an overdrive. Idler gears reverse the output direction without changing the ratio. Multiple gear stages multiply together, so a two-stage gearbox with ratios 3 and 4 has an overall ratio of 12.
Vehicle wheel speed and engine RPM
A car wheel turns at a rate determined by road speed and tyre circumference: wheel RPM = speed / (pi x tyre_diameter) x 60 (with speed in m/s). The engine turns faster by the overall drivetrain ratio, the product of the current gear ratio and the final drive (differential) ratio. This relationship is why fitting a larger-diameter tyre lowers cruise engine RPM for the same road speed, and why a lower (numerically higher) final drive ratio makes the engine rev higher. The calculator asks for the combined transmission x final-drive ratio; for a direct-drive axle that ratio is 1.
Machining spindle speed (the N = Vc x 1000 / pi x D formula)
Every cutting material has a recommended surface cutting speed in metres per minute (or surface feet per minute, SFM, in the US). Too fast and the cutting edge overheats and fails; too slow wastes time and can cause rubbing. From the recommended cutting speed and the tool or workpiece diameter, the required spindle RPM is N = (Vc x 1000) / (pi x D), where Vc is in m/min and D in mm. The factor of 1000 converts mm to m. Material presets here (aluminium, mild steel, stainless, brass, plastics, wood) are typical starting points for uncoated HSS tooling; carbide grades can run 3-5x faster.
Typical cutting speeds for common materials (HSS tooling)
| Material | Cutting speed (m/min) | Cutting speed (SFM) | Notes |
|---|---|---|---|
| Aluminium alloys | 200-400 | 650-1300 | Can go higher with sharp HSS |
| Mild steel | 20-40 | 65-130 | Reduce speed for heavy cuts |
| Stainless steel (304) | 15-25 | 50-80 | Use coolant; work-hardens quickly |
| Brass / bronze | 70-110 | 230-360 | Free-machining grades at higher end |
| Plastics (general) | 100-200 | 330-660 | Sharp tool, low heat to avoid melting |
| Wood (routing) | 400-800 | 1300-2600 | Router bits; chip evacuation critical |
| Cast iron | 20-40 | 65-130 | Dry cutting; brittle chips |
Use these as starting points. Carbide inserts can run 3-5x faster. Always consult the tool manufacturer data for exact values.
Frequently asked questions
How do I convert RPM to rad/s?
Multiply the RPM by 2pi and divide by 60: omega = RPM x 2pi / 60. For example, 1000 RPM = 1000 x 2pi / 60, which is approximately 104.72 rad/s.
What is the difference between angular velocity and surface speed?
Angular velocity (rad/s or deg/s) describes how fast the angle is changing and is the same everywhere on a rigid spinning body. Surface speed (m/s or ft/s) is how fast a specific point actually moves through space; it equals angular velocity multiplied by the radius to that point. Outer edges always move faster than inner ones at the same RPM.
How does a pulley belt drive change RPM?
The driven pulley RPM equals the driver RPM multiplied by the ratio of driver diameter to driven diameter, minus any belt slip. A 100 mm driver pulley at 1000 RPM connected to a 200 mm driven pulley produces 500 RPM before slip. With 2% V-belt slip the result is 490 RPM.
How do I calculate spindle RPM for machining?
Use the formula N = (Vc x 1000) / (pi x D), where Vc is the recommended cutting speed in m/min and D is the tool or workpiece diameter in mm. For a 25 mm end mill in aluminium at 300 m/min, N = (300 x 1000) / (pi x 25) = 3820 RPM.
How do I find RPM from vehicle speed and tyre size?
Convert speed to m/s, then divide by the tyre circumference (pi x outer diameter in metres) and multiply by 60 to get wheel RPM. Multiply wheel RPM by the overall gear ratio to get engine RPM. For example, 100 km/h with a 650 mm tyre gives wheel RPM = (100 / 3.6) / (pi x 0.65) x 60 = approximately 811 RPM.
How do gear teeth determine the speed ratio?
The speed ratio equals the number of driven teeth divided by the number of driver teeth. A 20-tooth driver meshing with a 60-tooth gear gives a 3:1 reduction, so 1450 RPM input becomes 483 RPM output, with output torque tripled (minus gear friction).