What this calculator does
Find out how fast something is turning, in rad/s, degrees per second or rpm. Angular velocity describes how quickly something spins. Instead of asking how many meters an object covers each second, it asks how much angle it sweeps through each second. It's written ω (the Greek letter omega). Its standard (SI) unit is the radian per second, where a radian is the angle that cuts off an arc as long as the radius. A wheel, a turntable, a planet and a fan blade can each be described by one number that applies to every point on them at once.
Several everyday measurements lead to the same ω. You can divide the angle turned by the time taken. You can divide the speed of a point on the rim by its distance from the center. Or you can multiply how many turns it makes per second by 2π. This calculator handles all four starting points. It gives the answer in radians per second, degrees per second, revolutions per minute and revolutions per second. It also shows the frequency and the time for one full turn.
Where it's used
Why this math matters outside the classroom:
- Engine speed. Mechanics convert an engine turning at 3,000 rpm to 3,000 × 2π ÷ 60, which is about 314 rad/s.
- Wheel and rim speed. A bike wheel of 0.35 m radius spinning at 20 rad/s carries its rim along at 20 × 0.35 = 7 m/s.
- Earth's rotation. Astronomy students find Earth turns 360° in about 24 hours, roughly 15 degrees per hour, which is why stars drift across the sky.
- Disk drives. A disk platter at 7,200 rpm makes 120 turns each second, so one rotation takes about 8.3 milliseconds.
Grade level and standards: Grades 11–12; Common Core State Standards for Mathematics HSF.TF.A.1. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Choose what you know: an angle and the time it took, a linear speed and radius, a frequency, or a period.
- Enter the values and pick their units, such as degrees or turns, seconds or milliseconds, km/h or m/s.
- Choose the unit you want for the answer.
- Read the angular velocity, its value in the other units, and the matching frequency and period.
What the inputs mean
- Angle turned (Δθ)
- The total angle swept in the time given. One full revolution is 360°, 2π radians or 1 turn, so 2.5 turns is 900°.
- Linear (tangential) speed (v)
- The speed of a point moving along the circle, measured along its path. Points farther from the center move faster at the same ω.
- Radius of the path (r)
- The distance from the center of rotation (the axis) to the point whose speed you entered.
- Frequency (f)
- The number of complete turns or cycles per second, in hertz (Hz). The angular frequency of a vibration is found the same way.
Formula and method
ω = Δθ ÷ Δt ω = v ÷ r ω = 2π × f ω = 2π ÷ T rpm = ω × 60 ÷ 2π
Here ω is the angular velocity, Δθ is the angle turned and Δt is the time it took. v is the speed of a point along its path and r is its distance from the axis. f is the frequency in turns per second, T is the period (the time for one turn), and rpm is revolutions per minute.
The first formula is the definition: average angular velocity is the angle turned divided by the time. The angle must be in radians for the answer to come out in rad/s. So degrees and turns are converted first (1° = π ÷ 180 rad and 1 turn = 2π rad).
The other three come from geometry. A point at radius r that turns through θ radians travels an arc of length r × θ. So its speed is v = ω × r. One revolution is 2π radians and lasts one period T, so ω = 2π ÷ T. Since the frequency is f = 1 ÷ T, ω = 2π × f as well.
Worked example
A door swings open through 90° in 2 seconds. What is its average angular velocity?
- Convert the angle: 90° × 0.0174532925 = 1.570796 rad.
- Divide by the time: ω = 1.570796 ÷ 2 = 0.785398 rad/s.
- In other units that's 45 °/s or 7.5 rpm. One full turn at this rate would take 8 s.
Result: The door turns at about 0.7854 rad/s (π/4 rad/s), which is 45 degrees per second.
Understanding the result
Every point on a solid spinning object shares the same angular velocity, but not the same straight-line speed. A point twice as far from the center moves twice as fast. That's why the outer edge of a merry-go-round feels so much quicker than the middle.
For a shaft or motor, rpm is the familiar unit. For physics formulas such as centripetal force, rotational kinetic energy or angular momentum, use rad/s.
Assumptions and limitations
- The angle-and-time mode gives the average angular velocity over that time. If the spin speeds up or slows down, the value at any one instant differs.
- The speed-and-radius mode assumes motion in a circle around a fixed axis, with v measured along the path.
- Results are sizes only. The direction of spin (clockwise or counterclockwise) isn't tracked.
- Angular acceleration, which describes a changing ω, isn't calculated.
- Rotation in three dimensions, where the axis itself moves, needs vector methods.
- Very small angles over very long times may be outside the range that can be calculated reliably.
Frequently asked questions
How do I convert rpm to rad/s?
Multiply by 2π and divide by 60, which is the same as multiplying by about 0.10472. A motor at 3,000 rpm turns at roughly 314.16 rad/s.
Is angular velocity the same as angular frequency?
They share the formula ω = 2π × f and the unit rad/s. Angular velocity usually describes something that's actually spinning. Angular frequency describes how fast any repeating process moves through its cycle, such as a vibration or an alternating current.
Why are radians used instead of degrees?
With radians, arc length is just radius × angle. So formulas like v = ω × r and a = ω² × r need no extra conversion factors. In degrees, every one of those formulas would carry a factor of π ÷ 180.
What is the angular velocity of the Earth?
Measured against the stars, the Earth turns once every 23 hours 56 minutes 4 seconds or so. That gives about 7.29 × 10⁻⁵ rad/s. Using a 24-hour solar day instead gives a very slightly smaller value.
How this calculator is checked
Method: Definition of angular velocity (ω = Δθ/Δt) with the circular-motion relations v = ωr and ω = 2πf = 2π/T. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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