Angular Displacement Calculator
Compute angular displacement Δθ and angular position θ: find the missing variable (θf, θ₀, ω, t, Δθ), convert radians ↔ degrees ↔ revolutions, and compute arc length s = rθ. The formula θ = θ₀ + ωt below only applies for constant ω.
Solver: find the missing variable
Angle converter: radians ↔ degrees ↔ revolutions
1 revolution = 360° = 2π radians. For Δθ, the "revolution" conversion result can be fractional or negative.
Angular velocity ω converter
Static ω converter. Need the frequency ↔ ω relationship? See hz-to-rads or the ω↔f tab in frequency-converter.
Time converter
Arc length s = r × θ (helper panel)
Helper tool for converting Δθ into linear distance along a circle. This is not a full arc-length calculator — just s = rθ with θ in radians.
Formulas
Δθ = θf − θiangular displacement = final angle minus initial angle
θf = θ₀ + ωt → Δθ = ωtonly for CONSTANT ω during interval t
s = r × θ (θ in radians)arc length = radius times angle
Worked examples (computed by the engine)
180° = 3.1415927 rad, θf = 210° = 3.6651914 rad0.5 rad/s = 28.64789 °/s−160° = −2.7925268 rad (negative = clockwise)1260° = 21.991149 rad6.2831853 m376.99112 rad/sHonesty notes
Scope limits (anti-cannibalization)
- This tool = angular position and displacement θ. Angular velocity ω as the main topic is a separate calculator (angular velocity).
- Not a frequency converter: the static f ↔ ω relationship is in hz-to-rads and frequency-converter.
- Not a torque calculator: shaft twist angle from torque (φ = TL/GJ) is in angle-of-twist.
- Not an angular acceleration calculator: α = Δω/Δt is in angular-acceleration-calculator.
- Not a full arc-length calculator: the Arc Length tab is just a helper panel for s = rθ.
FAQ
What is angular displacement?
Angular displacement, written Δθ (delta theta), is the change in angular position of a rotating object: final angle minus initial angle. Its SI unit is the radian, but degrees and revolutions are also common. Unlike distance traveled, Δθ can be negative and can exceed 360 degrees.
What is the angular displacement formula?
The basic formula is Δθ = θf − θi, final angle minus initial angle. If angular velocity ω is constant during time interval t, θf = θ₀ + ωt also holds, so Δθ = ωt. Example: θ₀ = 30°, ω = 45°/s, t = 4 s gives Δθ = 180° = 3.1415927 rad and θf = 210° = 3.6651914 rad.
What is the difference between angular displacement and angular velocity?
Angular displacement (θ) is angular POSITION or its change, while angular velocity (ω) is the RATE of angle change per unit time. This calculator computes θ displacement and position; to compute angular velocity ω as the main topic, use the separate angular velocity calculator.
Can angular displacement be negative?
Yes. The sign of Δθ indicates rotation direction: with the math convention (counterclockwise positive), negative Δθ means clockwise rotation. Example: θi = 120° and θf = −40° gives Δθ = −160° = −2.7925268 rad. A negative value is valid direction information, not an error.
What is the difference between angular displacement and revolution count?
Total revolutions = Δθ divided by 360°, and the value can be fractional or negative. This differs from the count of completed FULL revolutions. Example: Δθ = 450° equals 1.25 revolutions, but only 1 full revolution is actually completed. This calculator shows both so they are not confused.
When can the formula θ = θ₀ + ωt be used?
Only if angular velocity ω is CONSTANT throughout the entire time interval t. If ω changes (there is angular acceleration), this formula does not apply and the result is wrong. For changing ω, use the angular acceleration calculator or rotational motion integration methods.
How do you convert radians, degrees, and revolutions?
One full revolution = 360° = 2π radians. So 1° = π/180 radians and 1 radian = 180/π degrees. Example: 3.5 revolutions = 1260° = 21.991149 rad. This calculator converts all three both ways instantly.
How do you find angular velocity from angular displacement?
Use ω = Δθ / t = (θf − θ₀) / t, requiring t greater than 0. Example: θ₀ = 0.5 rad, θf = 2.0 rad, t = 3 s gives ω = 0.5 rad/s = 28.64789 °/s. Note this is the average ω over that interval and assumes constant ω.
How is angular displacement related to arc length?
Arc length s = r × θ, with θ in radians and r the path radius. Example: r = 2 m and Δθ = 180° (π radians) gives s = 6.2831853 m. The arc length panel in this calculator is just a helper; this is not a full arc-length calculator.
What units are used for angular velocity ω?
The SI unit is radians per second (rad/s). Other common units: degrees per second (°/s), revolutions per second (rev/s), and RPM (revolutions per minute). Example: 3600 RPM = 376.99112 rad/s. For frequency-to-ω conversion, see also the hz-to-rads calculator.
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