Velocity - Angular Converter
Convert angular velocity, how fast something rotates, between radians, degrees and revolutions per second, minute, hour or day. Pick two units, enter a value and read the result instantly.
Select two units to see the conversion factor.
This tool converts angular velocity (angle per time) only. It is not linear velocity (m/s), which also needs a radius, and it is not angular acceleration (angle per time squared).
Angular Velocity Formula
Every angular velocity unit is a fixed multiple of the radian per second. Result = value × (source factor ÷ target factor). The building blocks are exact: 1 revolution = 2π rad = 360°, 1 degree = π/180 rad, 1 minute = 60 s, 1 hour = 3,600 s and 1 day = 86,400 s. So 1 rev/min = 2π ÷ 60 rad/s, and 1 rev/min = 6 °/s.
What Is Angular Velocity?
Angular velocity tells you how quickly an object turns about an axis: the angle swept per unit of time, ω = Δθ ÷ Δt. It is the same for every point of a rigid rotating body, whether the point is near the axis or at the rim. A wheel turning at 300 rev/min has one angular velocity, but its rim moves much faster than its hub.
The SI unit is the radian per second (rad/s), because the radian is the natural angle unit in physics and engineering formulas. Everyday practice uses other units: revolutions per minute (rpm) for engines, motors, drives and turbines, degrees per second for gyroscopes, cameras and robot joints, and revolutions or radians per hour or day for slow rotations such as a clock hand or the Earth. Angular velocity also has a direction, given by the right-hand rule along the axis, so a negative value means rotation the other way.
Do not confuse it with related quantities. Angular velocity is angle per time. Linear (tangential) velocity is distance per time and is found with v = ω × r, but only when ω is in rad/s. Angular velocity is also not angular acceleration, the rate at which ω itself changes (Acceleration - Angular Converter). A rotation rate of 1 rev/s is a rotational frequency of 1 Hz, but ω = 2πf, so 1 Hz corresponds to 2π rad/s.
Supported Units
This converter supports the following 12 angular velocity units. The last column gives the value of each unit in the base unit, the radian per second. Radian-, degree- and revolution-based units are all converted through the exact relations 1 rev = 2π rad = 360°.
| Unit | Symbol | Value in rad/s |
|---|---|---|
| Radian per second | rad/s | 1 |
| Degree per second | °/s | 0.01745329252 |
| Revolution per minute (rpm) | rev/min | 0.1047197551 |
| Revolution per second | rev/s | 6.283185307 |
| Radian per minute | rad/min | 0.01666666667 |
| Radian per hour | rad/h | 0.0002777777778 |
| Radian per day | rad/d | 1.157407407e-5 |
| Degree per minute | °/min | 0.0002908882087 |
| Degree per hour | °/h | 4.848136811e-6 |
| Degree per day | °/d | 2.020057005e-7 |
| Revolution per hour | rev/h | 0.001745329252 |
| Revolution per day | rev/d | 7.272205217e-5 |
How to Use the Velocity - Angular Converter
- Type the angular velocity in the value box. Decimals and scientific notation such as 0.0025 or 2.5e-3 are accepted.
- Choose the From unit and the To unit.
- The result appears instantly, or press Convert Now. Very large or very small results are shown in scientific notation.
- Use Swap Units to reverse the conversion, Copy Result to copy the answer, open Show this value in all units for a full list, and Clear to start again.
Step-by-step by hand
- Find the size of each unit in the common base unit (rad/s): 1 rev/min = 0.1047197551 rad/s and 1 rad/s = 1 rad/s.
- Divide the source size by the target size to get the conversion factor: 0.1047197551 ÷ 1 = 0.1047197551, so 1 rev/min = 0.1047197551 rad/s.
- Multiply by your value: 3,000 rev/min × 0.1047197551 = 314.1592654 rad/s.
Angular velocity can be positive or negative (the sign shows the direction of rotation), so a minus sign is accepted. Zero is also valid.
Common Angular Velocity Conversions
- Engine at 3,000 rpm: 3,000 rev/min equals 314.1592654 rad/s, which is 50 rev/s or 18,000 °/s.
- Hard disk at 7,200 rpm: 7,200 rev/min equals 120 rev/s and 753.9822369 rad/s.
- Vinyl record at 33⅓ rpm: equals 3.490658504 rad/s.
- One turn per day: 1 rev/day equals 7.272205217e-5 rad/s. (The Earth turns slightly faster than this, about 7.292 × 10−5 rad/s, because a day relative to the stars is shorter than 24 hours.)
- Gyroscope reading: 90 °/s equals 1.570796327 rad/s and 15 rev/min.
- Back to SI: 100 rad/s equals 954.9296586 rev/min.
Revolution per Minute to Other Angular Velocity Units
Quick reference for frequently used values
| rev/min (rpm) | Radian/second | Degree/second | Revolution/second |
|---|---|---|---|
| 1 | 0.1047197551 | 6 | 0.01666666667 |
| 10 | 1.047197551 | 60 | 0.1666666667 |
| 60 | 6.283185307 | 360 | 1 |
| 100 | 10.47197551 | 600 | 1.666666667 |
| 500 | 52.35987756 | 3,000 | 8.333333333 |
| 1,500 | 157.0796327 | 9,000 | 25 |
| 3,000 | 314.1592654 | 18,000 | 50 |
| 7,200 | 753.9822369 | 43,200 | 120 |
Where and Why This Conversion Is Used
- Motors, engines and turbines: speeds are quoted in rpm, but power calculations (P = torque × ω) need rad/s.
- Mechanical design: shafts, gears, flywheels, fans and pumps, where rim speed v = ω × r needs ω in rad/s.
- Sensors and robotics: gyroscopes, IMUs, encoders and servo joints report degrees per second or radians per second.
- Data storage and media: disk and turntable speeds in rpm.
- Astronomy and geophysics: very slow rotations in radians or degrees per hour or day.
Limitations
- The converter changes units only. To get a linear speed you must multiply the angular velocity in rad/s by a radius.
- It treats every value as the magnitude (and sign) of rotation about one axis; it does not combine rotations about different axes.
Important Differences Between Related Measurements
- Angular velocity vs linear velocity: angular velocity is angle per time (rad/s) and is the same for every point on a rigid body. Linear velocity is distance per time and grows with radius: v = ω × r. A point 0.5 m from the axis of a shaft turning at 10 rad/s moves at 5 m/s.
- Angular velocity vs angular acceleration: angular velocity is how fast something spins. Angular acceleration is how quickly that spin changes, in rad/s². A flywheel can have a high angular velocity and zero angular acceleration.
- Angular velocity vs frequency: rotational frequency counts turns per second (Hz or rev/s). Angular velocity counts radians per second, so ω = 2π × f.
- Degrees vs radians in formulas: v = ω × r, a = α × r and P = τ × ω hold only with ω in rad/s. Convert rpm or °/s to rad/s first.
Each converter on this site changes units within one quantity only, and none of them mixes quantities. The related tools are: Velocity - Angular, Acceleration, Acceleration - Angular, Density, Specific Volume, Moment of Inertia, Moment of Force and Torque.