What this calculator does
Angular speed describes how fast something rotates, measured as the angle swept out per unit of time rather than a straight-line distance covered. It is the rotational equivalent of ordinary speed, and it comes up wherever something spins: wheels, motors, turntables, planets and gears.
There are three common starting points for finding it, and this calculator covers all three: an angle turned over a measured time, a point’s linear speed at a known distance from the centre of rotation, or a rotational speed already expressed in revolutions per minute. Whichever one you start from, the underlying angular speed is the same physical quantity, just reached by a different route.
The formula
For an angle and time, divide the angle, converted to radians, by the time taken. For a linear speed and radius, divide the linear speed by the radius, since a point further from the centre must travel faster to complete the same rotation in the same time. For RPM, multiply the revolutions per minute by 2π to convert revolutions to radians, then divide by 60 to convert minutes to seconds.
| Term | Meaning |
|---|---|
| Angular speed (ω) | The rate of rotation, most naturally expressed in radians per second. |
| ω = θ / t | Angle turned divided by time taken. |
| ω = v / r | Linear speed of a point divided by its distance from the centre of rotation. |
| ω = 2πN / 60 | Rotational speed N in revolutions per minute, converted to radians per second. |
The inputs explained
| Field | What to enter |
|---|---|
| Known values | Which two values you already know: an angle and a time, a linear speed and a radius, or a rotational speed in RPM. |
| Angle turned in degrees (mode 1), linear speed in m/s (mode 2), or RPM (mode 3) | The angle in degrees, the linear speed in metres per second, or the RPM figure, depending on the mode selected above. |
| Time taken in seconds (mode 1), or radius in metres (mode 2); not used in RPM mode | The time in seconds, or the radius in metres, depending on the mode selected above. Not used in RPM mode. |
When to use it
Working from a stopwatch measurement
Timing how long an object takes to turn through a known angle, such as a quarter turn, gives the angle-and-time route to angular speed directly.
Working from a point on a spinning disc
If the speed of a marked point on a wheel or disc is known, along with how far that point sits from the centre, the linear-speed-and-radius route finds the disc’s angular speed without needing a separate timing measurement.
Converting a motor or turntable spec sheet
Motors and turntables are usually rated in RPM. Converting that figure to radians per second puts it on the same footing as other rotational-motion formulas, most of which are written in terms of radians.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How angular speed changes with rotational speed in RPM
A range of common rotational speeds, converted to angular speed.
| Rotational speed | Angular speed (ω) | Equivalent rotational speed |
|---|---|---|
| 33 RPM | 3.456 rad/s | 33.00 RPM |
| 45 RPM | 4.712 rad/s | 45.00 RPM |
| 78 RPM | 8.168 rad/s | 78.00 RPM |
| 300 RPM | 31.416 rad/s | 300.00 RPM |
| 1000 RPM | 104.720 rad/s | 1,000.00 RPM |
| 3000 RPM | 314.159 rad/s | 3,000.00 RPM |
How angular speed changes with radius, at a fixed linear speed
A point moving at a constant 10 m/s, at a range of distances from the centre of rotation.
| Radius | Angular speed (ω) | Angular speed |
|---|---|---|
| 0.5 m | 20.000 rad/s | 1,145.92 °/s |
| 1 m | 10.000 rad/s | 572.96 °/s |
| 2 m | 5.000 rad/s | 286.48 °/s |
| 5 m | 2.000 rad/s | 114.59 °/s |
| 10 m | 1.000 rad/s | 57.30 °/s |
| 20 m | 0.5000 rad/s | 28.65 °/s |
Questions
What is the difference between angular speed and linear speed?
Linear speed measures distance covered per unit time along the path of motion. Angular speed measures angle swept per unit time. For circular motion the two are linked by the radius: linear speed equals angular speed multiplied by radius.
Why are there three different formulas for the same quantity?
They are three different starting points for measuring the same physical rotation, chosen based on what is easiest to measure in a given situation, a timed angle, a linear speed at a known radius, or a rated RPM figure. All three give the same angular speed for the same actual rotation.
Is angular speed the same as angular velocity?
In everyday usage they are often used interchangeably, but strictly speaking angular velocity is a vector with a direction (the axis of rotation and a sign for the rotation sense), while angular speed is just its magnitude, the number without the direction.
How is this different from the RPM converter?
The RPM converter starts only from a known RPM figure and converts it to other units. This calculator also accepts an angle-and-time measurement or a linear-speed-and-radius measurement as the starting point, for situations where RPM was never directly known.
If you already have a rotational speed in RPM and just need it converted to other units, see the RPM converter.