Quick answer
Angular velocity ω = 2π × rpm ÷ 60 = θ ÷ t = 2πf. 3,000 rpm is 314.159 rad/s, a frequency of 50 Hz and 18,000 degrees per second.
Choose the quantity you know (rpm, frequency, or angle and time) and enter it. The calculator returns angular velocity in several units.
Related tools: Angular Acceleration Calculator: Angular acceleration from a change in angular velocity. Torque Calculator: Torque from force, lever arm and angle. Water Velocity Calculator: Water velocity in a pipe from flow and inside diameter.
How to use the Angular Velocity Calculator
- Choose starting from: From revolutions per minute, From angle and time and From frequency.
- Optionally, enter revolutions per minute (above 0).
- Optionally, enter angle turned (rad).
- Optionally, enter time (s) (above 0).
- Optionally, enter frequency (Hz) (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Starting from | Choose one: From revolutions per minute; From angle and time; From frequency | Starts at From revolutions per minute |
| Revolutions per minute (optional) | A number | > 0 |
| Angle turned (optional) | A number, in rad | — |
| Time (optional) | A number, in s | > 0 |
| Frequency (optional) | A number, in Hz | > 0 |
Formula and method
Formula
ω = 2π × rpm ÷ 60 = θ ÷ t = 2πf
IEEE-754 double precision; shown to the decimals indicated.
Shown in rad/s, rpm, Hz and degrees per second.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Starting from = From revolutions per minute; Revolutions per minute = 3000.
| Result | Value |
|---|---|
| Angular velocity | 314.159265 rad/s |
| Revolutions per minute | 3,000 |
| Frequency | 50 Hz |
| Degrees per second | 18,000 |
Example 2
Inputs: Starting from = From angle and time; Angle turned = 6.2832; Time = 2.
| Result | Value |
|---|---|
| Angular velocity | 3.1416 rad/s |
| Revolutions per minute | 30.0001 |
| Frequency | 0.500001 Hz |
| Degrees per second | 180.0004 |
The first line is the headline result (angular velocity). Below it the angular Velocity Calculator lists revolutions per minute, frequency and degrees per second, which describe the same case from other angles. IEEE-754 double precision; shown to the decimals indicated.
Conversions
3,000 ÷ 60 = 50 revolutions per second (Hz); each is 2π radians, so ω = 314.16 rad/s. Each revolution is 360°, so 18,000°/s.
Linear speed
The speed at radius r is v = ωr. A point 0.1 m from the axis moves at 31.4 m/s. Only the fields for the chosen mode are used.
How the result changes with revolutions per minute
Holding every other input at the first worked example, this table changes only revolutions per minute. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Revolutions per minute | Angular velocity | Revolutions per minute | Frequency |
|---|---|---|---|
| 1500 | 157.079633 rad/s | 1,500 | 25 Hz |
| 3000 | 314.159265 rad/s | 3,000 | 50 Hz |
| 6000 | 628.318531 rad/s | 6,000 | 100 Hz |
| 12000 | 1,256.637061 rad/s | 12,000 | 200 Hz |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Revolutions per minute: left empty shows "Enter the revolutions per minute." beside that field and no result.
- Time: left empty shows "Enter the time." beside that field and no result.
Choosing between the Angular Velocity Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Angular Acceleration Calculator](/science/angular-acceleration-calculator/) | initial angular velocity, final angular velocity and time | Angular acceleration |
| [Torque Calculator](/science/torque-calculator/) | lever arm, force and angle between force and lever | Torque |
| [Water Velocity Calculator](/science/water-velocity-calculator/) | flow and inside diameter | Velocity |
| [Escape Velocity Calculator](/science/escape-velocity-calculator/) | body | Escape velocity |
| [Terminal Velocity Calculator](/science/terminal-velocity-calculator/) | mass, frontal area and drag coefficient | Terminal velocity |
Background: gravity, rotation and optics
Gravity and rotation calculations use a few core relations: a circular orbit has speed v = √(GM ÷ r), torque equals moment of inertia times angular acceleration (τ = Iα), and a thin lens obeys 1/f = 1/dₒ + 1/dᵢ. The moment of inertia depends on both the shape and the axis of rotation.
Use SI units (metres, kilograms, seconds) and check the sign conventions of the formula before interpreting a negative result.
Assumptions and limitations
- Only the fields for the chosen mode are used
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for revolutions per minute and time to confirm this.
Frequently asked questions
What is 1 rad/s in rpm?
1 rad/s is 60 ÷ 2π ≈ 9.549 rpm.
What is the angular velocity of Earth's rotation?
About 7.29 × 10⁻⁵ rad/s.
What is frequency?
Revolutions per second.
What does the Angular Velocity Calculator calculate?
Angular velocity from rpm, frequency, or an angle and time. It returns Angular velocity, Revolutions per minute, Frequency and Degrees per second.
What formula does the Angular Velocity Calculator use?
ω = 2π × rpm ÷ 60 = θ ÷ t = 2πf. Shown in rad/s, rpm, Hz and degrees per second.
What do I need to enter in the angular Velocity Calculator?
You enter Starting from. Optional: Revolutions per minute, Angle turned (rad), Time (s) and Frequency (Hz).
What does the “Starting from” option change?
It sets how the angular Velocity is worked out. The choices are From revolutions per minute, From angle and time and From frequency.
What is an example of a angular Velocity result?
With starting from From revolutions per minute and revolutions per minute 3000, the angular velocity is 314.159265 rad/s.
Related calculators
- Angular Acceleration Calculator — Angular acceleration from a change in angular velocity.
- Torque Calculator — Torque from force, lever arm and angle.
- Water Velocity Calculator — Water velocity in a pipe from flow and inside diameter.
- Escape Velocity Calculator — Speed needed to escape a body's gravity from its surface.
- Terminal Velocity Calculator — Terminal speed of a falling object from mass, area and drag coefficient.
Sources
- OpenStax, University Physics Volume 1 (Rice University): mechanics
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.