What is Angular Velocity? A Complete Guide
A practical guide to angular velocity. Learn what it is, how it differs from linear velocity, and why it is universally measured in radians per second.
Read articleRPM to Radians Per Second Converter
We have all been there: staring at a motor's RPM spec when your equations demand rad/s. This calculator bridges that gap instantly. Convert RPM, rad/s, Hertz, and Degrees per Second with zero rounding errors. Add an optional radius to grab the surface linear velocity while you are at it.
Last updated:
How it works
- 1
Select Your Starting Unit
Whether you have a datasheet in RPM or a physics problem in rad/s, pick the unit you already know. No need to overthink it.
- 2
Drop In The Numbers
Type your figure. Add an optional radius in meters if you want to skip a step and get the linear surface velocity (m/s, km/h, mph) immediately.
- 3
Get The Full Picture
We handle the math. Grab your perfectly converted angular velocity, period in seconds, and a real-world benchmark so you know if your answer makes sense.
The Math: RPM to Radians Per Second
Let's be real: RPM is great for mechanics, but it's a nightmare for calculus. When you need to plug rotational speed into an equation, you almost always need rad/s. The math to get there is actually straightforward. One full revolution is exactly 2π radians, and one minute is 60 seconds. So, multiply your RPM by 2π and divide by 60. This reduces cleanly to multiplying by π/30. Here is the trench truth: do not just multiply by 0.10472. That is a rounded decimal, and using it will slowly corrupt your answers if you chain it through multiple equations. Keep the π/30 to keep your math exact.
rad/s = RPM × 2π ÷ 60
= RPM × π/30
RPM = rad/s × 60 ÷ 2π
= rad/s × 30/π
Related:
Hz = RPM ÷ 60
°/s = rad/s × 180/π
Period = 60 ÷ RPM (seconds)Example
Let's look at the numbers. Say you need to convert a 3,000 RPM motor to rad/s: 3,000 × π/30 = 100π, which is roughly 314.159 rad/s. For the related units, that is 3,000 ÷ 60 = 50 Hz, 314.159 × 180/π = 18,000 °/s, and a rotation period of 60 ÷ 3,000 = 0.02 seconds per revolution.
Worked examples
| Scenario | Calculation | Result |
|---|---|---|
| Machine A — car engine at 3,000 RPM → rad/s | 3,000 × π/30 | 314.159265 rad/s · 50 Hz · 18,000 °/s · 0.02 s/rev |
| Motor B — electric motor at 1,500 RPM → rad/s | 1,500 × π/30 | 157.079633 rad/s · 25 Hz · 9,000 °/s · 0.04 s/rev |
| Drive C — hard drive at 7,200 RPM → rad/s + Hz | 7,200 × π/30 (and 7,200 ÷ 60) | 753.982237 rad/s · 120 Hz · 43,200 °/s |
| Tool D — angle grinder at 11,000 RPM → rad/s | 11,000 × π/30 | 1,151.917306 rad/s · 183.33 Hz · 66,000 °/s |
| Reverse — Motor E at 100 rad/s → RPM | 100 × 30/π | 954.929659 RPM · 15.915 Hz · 5,729.578 °/s |
Conversion reference
| RPM | rad/s | Hz | °/s | Period (s) |
|---|---|---|---|---|
| 1 | 0.104720 | 0.016667 | 6 | 60 |
| 10 | 1.047198 | 0.166667 | 60 | 6 |
| 60 | 6.283185 | 1 | 360 | 1 |
| 100 | 10.471976 | 1.666667 | 600 | 0.6 |
| 120 | 12.566371 | 2 | 720 | 0.5 |
| 500 | 52.359878 | 8.333333 | 3,000 | 0.12 |
| 600 | 62.831853 | 10 | 3,600 | 0.1 |
| 1,000 | 104.719755 | 16.666667 | 6,000 | 0.06 |
| 1,500 | 157.079633 | 25 | 9,000 | 0.04 |
| 2,000 | 209.439510 | 33.333333 | 12,000 | 0.03 |
| 3,000 | 314.159265 | 50 | 18,000 | 0.02 |
| 5,000 | 523.598776 | 83.333333 | 30,000 | 0.012 |
| 7,200 | 753.982237 | 120 | 43,200 | 0.008333 |
| 10,000 | 1,047.197551 | 166.666667 | 60,000 | 0.006 |
Quick facts
- The exact multiplier is π/30 rad/s per RPM. Do not rely on the rounded 0.10472 if you need high precision.
- 1 Hz means one full revolution per second, which is exactly 2π rad/s (approx. 6.28 rad/s).
- A typical car engine redlining at 8,000 RPM is spinning at a blistering 837.8 rad/s.
- Going backward? The exact reverse factor is 30/π. Multiply rad/s by that to get your RPM back.
- A standard 7,200 RPM hard drive hums along at exactly 120 Hz, or roughly 754 rad/s.
From the blog
View all →radians per second to rpm8 min
·radians per second to rpm
Radians Per Second to RPM: The Ultimate Reverse Guide
Convert radians per second to RPM flawlessly. Learn the exact 30/π reverse multiplier, avoid decimal drift, understand motor sizing, and use our free calculator.
Read articlerpm to radians per second10 min
·rpm to radians per second
RPM to Radians Per Second: The Ultimate Conversion Guide
Master the rpm to radians per second conversion. Get the exact formula, avoid critical rounding errors, understand the calculus, and use our free calculator.
Read articleFrequently asked questions
We have all been there, overcomplicating a simple conversion. The math is straightforward: multiply your RPM value by π/30. That's it. For example, a 2,000 RPM motor: 2,000 × π/30 ≈ 209.44 rad/s. Skip the math anxiety and use our rpm to radians per second calculator at the top of the page.
If you are asking how do you convert rpm to radians per second manually, it is quite simple. You multiply your RPM value by the fraction π/30. This ensures your conversion stays exactly precise for any engineering or physics equations you are working on.
The exact rpm to radians per second formula is: rad/s = RPM × (π/30). Multiply your RPM by π, then divide by 30 to get precisely the number of radians per second. This avoids the compounding rounding errors you get from using a rounded decimal.
1 rpm to radians per second is exactly π/30 rad/s. As a decimal, this is approximately 0.10472 radians per second. It means that an object rotating at 1 RPM will sweep through roughly 0.10472 radians every single second.
The radians per second to rpm formula is RPM = rad/s × 30/π. If you want to go the other way—rpm to angular velocity radians per second—the formula is rad/s = RPM × π/30. Multiply rad/s by roughly 9.5493 and you have your RPM.
Think of it this way: every revolution is 2π radians. Multiplying your RPM by 2π converts revolutions into radians. Then, dividing by 60 converts your 'per-minute' rate into a 'per-second' rate. Combine them: 2π ÷ 60 neatly reduces to π/30. It's that simple.
Angular velocity is just the formal physics way of asking, 'how fast is this thing spinning?' In SI units, it is measured in radians per second (rad/s). An object turning at 1 rad/s sweeps through one radian (about 57.3 degrees) every single second.
Just multiply your rad/s value by 30/π (which is about 9.5493). This perfectly reverses the π/30 factor. For example, if you have 100 rad/s, 100 × 30/π gives you roughly 954.93 RPM. That is exactly how to convert radians per second to rpm.
120 rpm to radians per second is exactly 4π rad/s, which equals about 12.566 rad/s. This is an incredibly common conversion for low-speed synchronous motors, so it's a good one to memorize.
They measure the exact same thing, just for different audiences. RPM counts whole revolutions per minute and is perfect for mechanics reading a motor datasheet. rad/s measures the swept angle per second, which is what the math actually demands when doing calculus or physics equations.
Divide your RPM by 60. One hertz is simply one revolution per second, and there are 60 seconds in a minute. So, a 3,000 RPM motor runs at 3,000 ÷ 60 = 50 Hz.
The period is how long it takes to complete a single revolution. Just divide 60 by your RPM. At 3,000 RPM, the period is 60 ÷ 3,000 = 0.02 seconds per revolution. Fast and easy.
You need two things: your RPM and the radius of the rotating object. First, convert your RPM to rad/s by multiplying by π/30. Then, multiply that result by your radius in meters. The math is v = r × ω. Our calculator does this for you automatically if you drop in a radius.