Angular Velocity Calculator — Rad/s, RPM & More Instantly

Calculate angular velocity from angle & time, linear velocity & radius, RPM, frequency, or period. Free online tool with step-by-step formula breakdown and automatic unit conversions.

Verified Formulas 5 Calculation Modes Privacy First

Angular Velocity Calculator

Select a calculation mode and enter your known values to compute angular velocity in rad/s with automatic unit conversions.

Select a mode, enter values, and click Calculate Angular Velocity to see the result.

Angular Velocity Formulas Explained

Angular velocity (ω) measures how fast an object rotates or revolves around an axis, expressed in radians per second (rad/s). Choose the formula that matches your known values:

ω = Δθ / Δt — from angle and time
ω = v / r — from linear velocity and radius
ω = RPM × 2π / 60 — from revolutions per minute
ω = 2πf — from frequency in hertz
ω = 2π / T — from period in seconds

Variable Definitions

  • ω (omega) — Angular velocity in radians per second (rad/s)
  • Δθ — Angular displacement in radians
  • Δt — Time interval in seconds
  • v — Linear (tangential) velocity in meters per second (m/s)
  • r — Radius or distance from the axis of rotation in meters (m)
  • RPM — Revolutions per minute
  • f — Rotational frequency in hertz (Hz)
  • T — Period (time for one full revolution) in seconds

How to Calculate Angular Velocity

Follow these steps to calculate angular velocity using any of the five supported methods:

  1. Identify your known values — Determine whether you have angle & time, linear velocity & radius, RPM, frequency, or period data.
  2. Select the correct formula — Choose ω = Δθ/Δt, ω = v/r, ω = RPM×2π/60, ω = 2πf, or ω = 2π/T based on your inputs.
  3. Convert angle to radians if needed — If using degrees, multiply by π/180. If using revolutions, multiply by 2π.
  4. Plug in values and compute — Perform the division or multiplication to obtain ω in rad/s.
  5. Verify with unit conversions — Check your result against RPM (ω×60/2π), °/s (ω×180/π), or Hz (ω/2π).

Angular Velocity Calculation Examples

Example 1: From Angle & Time

A wheel rotates through 2π radians (one full revolution) in 0.5 seconds. Find the angular velocity.

ω = Δθ / Δt = 2π / 0.5 = 4π ≈ 12.57 rad/s
In RPM: 12.57 × 60 / 2π = 120 RPM

Example 2: From Linear Velocity & Radius

A point on a spinning disk of radius 0.3 m moves at 6 m/s. What is the angular velocity?

ω = v / r = 6 / 0.3 = 20 rad/s
In RPM: 20 × 60 / 2π ≈ 191 RPM

Example 3: From RPM

An engine runs at 3600 RPM. Calculate its angular velocity in rad/s.

ω = 3600 × 2π / 60 = 3600 × π / 30 ≈ 376.99 rad/s

Example 4: From Frequency

An AC motor operates at 60 Hz. Find the angular velocity.

ω = 2πf = 2π × 60 ≈ 376.99 rad/s

Real-World Angular Velocity Applications

  • Automotive Engineering: Converting engine RPM to rad/s for drivetrain and torque calculations.
  • Industrial Machinery: Specifying rotational speeds for lathes, milling machines, and conveyor belt drives.
  • Wind Turbine Design: Calculating blade tip speed from rotational velocity for aerodynamic efficiency.
  • Astronomy: Measuring planetary and stellar rotation rates in rad/s or degrees per day.
  • Sports Science: Analyzing spin rates of balls in baseball, tennis, and golf for performance optimization.
  • Gyroscope & Navigation: Sensing angular velocity for stabilization systems in drones and spacecraft.
  • HVAC Systems: Specifying fan and blower rotational speeds for airflow calculations.

People Also Ask

Angular velocity (ω) can be calculated using several formulas: ω = Δθ/Δt (from angle and time), ω = v/r (from linear velocity and radius), ω = 2πf (from frequency), ω = 2π/T (from period), and ω = RPM × 2π/60 (from RPM). The SI unit is radians per second (rad/s).
Multiply RPM by 2π and divide by 60: ω = RPM × 2π / 60. For example, 1200 RPM equals 1200 × 2π / 60 = 125.66 rad/s. One revolution equals 2π radians, and one minute equals 60 seconds, so this conversion is exact.
Angular velocity (ω) measures rotational speed in radians per second, describing how fast an object rotates around an axis. Linear velocity (v) measures straight-line speed in meters per second along a tangent. They are related by v = ω × r, where r is the radius from the axis.
The SI unit of angular velocity is radians per second (rad/s). Other common units include RPM, degrees per second (°/s), and hertz (Hz). One rad/s equals approximately 9.549 RPM or 57.296 degrees per second.
For a fixed linear velocity, angular velocity decreases as radius increases (ω = v/r). A point farther from the axis traces a larger circle, so it completes fewer radians per second at the same linear speed. Conversely, for fixed ω, linear velocity increases with radius.

Frequently Asked Questions

The calculator accepts angle in radians, degrees, or revolutions; linear velocity in m/s; radius in meters; RPM; frequency in Hz; and period in seconds. Results display in rad/s with automatic conversions to RPM, °/s, and Hz.
Yes. Angular velocity is essential for gear calculations. Use the RPM mode to input gear speeds, or the velocity & radius mode to relate linear and rotational motion in gear trains. For comprehensive gear ratio calculations, see our Gear Ratio Calculator.
Angular speed is a scalar quantity that measures how fast an object rotates without regard to direction. Angular velocity is a vector quantity that includes both magnitude and direction of rotation (clockwise or counterclockwise). This calculator computes the magnitude.
Radians per second (rad/s) is the SI standard for angular velocity because radians are dimensionless, making equations like v = ωr and a = ω²r dimensionally consistent without conversion factors. It simplifies physics and engineering calculations.
Yes. By convention, positive angular velocity indicates counterclockwise rotation, and negative indicates clockwise rotation. This calculator accepts negative inputs and preserves the sign in results, which is useful for vector-based mechanics problems.
All conversions use exact mathematical relationships: 1 revolution = 2π radians, 1 minute = 60 seconds, 1 Hz = 1 cycle/second = 2π rad/s. Results display with 6 significant figures of precision, suitable for engineering and academic use.

Angular Velocity Glossary

Angular Velocity (ω)

Rate of rotation around an axis, measured in rad/s. Equals angular displacement divided by time.

Radian (rad)

The SI unit of angle. One radian is the angle subtended by an arc equal to the radius. 2π rad = 360°.

RPM

Revolutions per minute, a common unit for rotational speed in machinery. 1 RPM = 2π/60 rad/s.

Tangential Velocity

The linear speed of a point on a rotating object, tangent to its circular path. v = ω × r.

Frequency (f)

Number of complete revolutions or cycles per second, measured in hertz (Hz). ω = 2πf.

Period (T)

Time required for one complete revolution or cycle, measured in seconds. T = 1/f = 2π/ω.

Centripetal Acceleration

Acceleration directed toward the center of rotation. a = ω²r = v²/r.

Moment of Inertia

Rotational analog of mass; resistance to changes in angular velocity. Depends on mass distribution about the axis.

Editorial Review & Methodology

This angular velocity calculator was built and reviewed by the NumbrWiz Editorial Team. All formulas are verified against standard physics and engineering references, including university-level dynamics textbooks and ISO 80000-3 standards for rotational quantities.

  • Formula verification: Cross-checked against Serway & Jewett's Physics for Scientists and Engineers and the NIST Guide to SI Units.
  • Edge case testing: Tested with zero-radius limits, negative values, extremely small periods, and high-RPM scenarios.
  • Unit conversion accuracy: All conversion factors use exact mathematical constants (2π, 60, etc.) with floating-point precision to 6 significant figures.
  • UX review: Five calculation modes designed to match common physics and engineering problem-solving workflows.

Transparency note: All calculations run client-side in your browser. No data is ever collected, stored, or transmitted. Results are for educational and engineering reference purposes; verify critical design calculations independently.

Page last reviewed: May 2026 · NumbrWiz Editorial Team