Rads to RPM: The Exact Formula and How to Convert Angular Velocity
Convert rad/s to RPM using RPM = rad/s × 30/π. Here is the full derivation, the exact factor, worked examples, and a reference table for common motor speeds.
Read articleRads to RPM Converter
Convert radians per second to RPM or RPM to rad/s instantly — complete with degrees per second, hertz, rotation period, and the nearest real-world machine benchmark.
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How it works
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Pick rad/s → RPM (the default) or switch to RPM → rad/s for the reverse conversion.
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Type any angular velocity, or choose a real-world example to load its rad/s equivalent automatically.
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See RPM, degrees per second, hertz, revolutions per hour, rotation period, and the closest real-world benchmark — all at once.
How to Convert Radians per Second to RPM
Physics, control systems, and simulation tools express angular velocity in radians per second because rad/s is the SI unit that calculus and rotational dynamics are built on. Motors, engines, tachometers, and mechanical datasheets almost always quote RPM, so switching between the two is a daily task in robotics, motor control, automotive engineering, and dynamics. The conversion is exact, not an approximation: one revolution is 2π radians and RPM counts revolutions per minute, so multiplying rad/s by 60 ÷ 2π — which simplifies to 30/π (≈ 9.5492965…) — gives RPM. That factor is irrational because π is irrational, so using 30/Math.PI in code keeps full floating-point precision. Going the other way, RPM × π/30 returns rad/s, while Hz = rad/s ÷ 2π, degrees per second = rad/s × 180/π, and the rotation period in seconds = 2π ÷ rad/s = 60 ÷ RPM.
RPM = rad/s × 60 ÷ 2π
= rad/s × 30/π
rad/s = RPM × 2π ÷ 60
= RPM × π/30
Related:
Hz = rad/s ÷ 2π
°/s = rad/s × 180/π
Period = 2π ÷ rad/s (seconds)Example
Convert 100 rad/s to RPM: RPM = 100 × 30/π = 3000/π ≈ 954.93 RPM. Alongside it, Hz = 100 ÷ 2π ≈ 15.92 Hz, °/s = 100 × 180/π ≈ 5,729.6 °/s, and the period = 2π ÷ 100 ≈ 0.0628 s per revolution.
Worked examples
| Scenario | Calculation | Result |
|---|---|---|
| Motor A — electric motor at 100 rad/s | 100 × 30/π | 954.93 RPM · 15.92 Hz · 5,729.6 °/s · 0.0628 s/rev |
| Servo B — servo drive at 200 rad/s | 200 × 30/π | 1,909.86 RPM · 31.83 Hz · 11,459.2 °/s · 0.0314 s/rev |
| Machine C — 52.36 rad/s (≈ 500 RPM) | 52.36 × 30/π | 500.00 RPM · 8.33 Hz · 3,000 °/s · 0.12 s/rev |
| Engine D — car engine at 3,000 RPM | 3000 × π/30 | 314.16 rad/s · 50 Hz · 18,000 °/s · 0.02 s/rev |
| Drive E — hard drive at 7,200 RPM | 7200 × π/30 | 753.98 rad/s · 120 Hz · 43,200 °/s · 0.00833 s/rev |
Conversion reference
| rad/s | RPM | Hz | °/s | Period (s) |
|---|---|---|---|---|
| 1 | 9.5493 | 0.1592 | 57.2958 | 6.2832 |
| 5 | 47.7465 | 0.7958 | 286.4789 | 1.2566 |
| 10 | 95.4930 | 1.5915 | 572.9578 | 0.6283 |
| 20 | 190.9859 | 3.1831 | 1,145.9156 | 0.3142 |
| 50 | 477.4648 | 7.9577 | 2,864.7890 | 0.1257 |
| 100 | 954.9297 | 15.9155 | 5,729.5780 | 0.0628 |
| 200 | 1,909.8593 | 31.8310 | 11,459.1559 | 0.0314 |
| 314.159 | 2,999.9975 | 50.0000 | 17,999.9848 | 0.0200 |
| 500 | 4,774.6483 | 79.5775 | 28,647.8898 | 0.0126 |
| 1000 | 9,549.2966 | 159.1549 | 57,295.7795 | 0.0063 |
Quick facts
- 1 rad/s = 30/π RPM = 9.5492965… RPM — an irrational number, because π is irrational.
- 1 RPM = π/30 rad/s = 0.10471975… rad/s — the exact reciprocal of the rad/s → RPM factor.
- A 1,500 RPM motor (a 4-pole machine on 50 Hz AC) spins at exactly 50π ≈ 157.08 rad/s.
- Hertz ties the units together: Hz = RPM ÷ 60 = rad/s ÷ 2π, so 3,000 RPM = 50 Hz = 100π rad/s.
- Multiplying by 30/π and by π/30 are exact inverses — convert and convert back and you land on the same number.
From the blog
View all →Frequently asked questions
Multiply the angular velocity in radians per second by 30/π (about 9.549). For example, 100 rad/s × 30/π ≈ 954.93 RPM. The factor comes from 60 seconds per minute divided by 2π radians per revolution.
RPM = rad/s × 60 ÷ 2π, which simplifies to RPM = rad/s × 30/π. To reverse it, use rad/s = RPM × π/30. Both factors are exact, so the conversion loses no information.
One revolution equals 2π radians and RPM counts revolutions per minute, so you scale rad/s by 60 ÷ 2π. The 60 and the 2 cancel to leave 30/π ≈ 9.5492965…, an irrational number because π itself is irrational.
Angular velocity measures how fast something rotates. In radians per second it states how many radians of angle are swept each second, where a full turn is 2π radians. It is the SI unit used throughout physics and control theory.
Multiply RPM by π/30 (about 0.10472). For example, 3,000 RPM × π/30 ≈ 314.16 rad/s. This is the exact inverse of the rad/s → RPM conversion.
Both describe rotational speed, but rad/s measures swept angle per second in SI units, while RPM counts whole revolutions per minute. rad/s suits equations and calculus; RPM suits motors, gauges, and datasheets.
Divide by 2π: Hz = rad/s ÷ 2π. Hertz is revolutions (or cycles) per second, so 100 rad/s ÷ 2π ≈ 15.92 Hz. Equivalently, Hz = RPM ÷ 60.
The period is the time for one full revolution: Period = 2π ÷ rad/s = 60 ÷ RPM seconds. At 100 rad/s the period is 2π ÷ 100 ≈ 0.0628 s. A value of 0 rad/s means the object is not rotating, so the period is infinite.