Angular Displacement Calculator

Related Calculators

Calculation method: The formula path the calculator uses, such as start/end angles, arc length and radius, angular velocity and time, or angular acceleration with time.

Angular displacement (selected unit): The change in angle from the starting position to the ending position. A positive value means rotation in the positive direction, and a negative value means the opposite direction.[2]

Start angle (selected unit) / End angle (selected unit): The beginning and ending angle values used to find the angle change.

Angle difference policy (start/end method): A rule for how the calculator handles crossing 0 degrees or 2 pi radians, such as raw difference or the smallest wrapped change.

Arc length traveled (meters): The distance moved along the circle, not straight across it.

Radius (meters): The distance from the center of the circle to the moving object.

Angular velocity (rad/s): How quickly angle changes with time, measured in radians per second.[1]

Angular acceleration (rad/s^2): How quickly angular velocity changes with time.

Angular displacement (radians): The same angle change shown in radians, which is the form many physics formulas use.

Angular displacement (revolutions): The angle change written as full turns, where 1 revolution equals 360 degrees.

Mistake: Entering 350 and 10 in Start angle (selected unit) and End angle (selected unit) but leaving Angle difference policy (start/end method) on raw when you wanted the short way around.
Fix: Change Angle difference policy (start/end method) to a wrapped option, then check Angular displacement (selected unit) again.

Mistake: Typing straight-line distance into Arc length traveled (meters) instead of the distance along the circle.
Fix: Use the path length along the rim for Arc length traveled (meters) , not the chord across the circle.

Mistake: Entering 0 or a negative number for Radius (meters) in the arc-length method.
Fix: Enter Radius (meters) as a number greater than 0 so the formula can work.

Mistake: Typing 90 into Angular velocity (rad/s) when your value is really 90 deg/s.
Fix: Open Angular velocity unit (input) in Advanced options and switch it to deg/s, or convert your value to rad/s first.

Mistake: Using the acceleration method but forgetting that Initial angular velocity (rad/s) and Angular acceleration (rad/s^2) can be negative when rotation is in the opposite direction.
Fix: Keep the sign on Initial angular velocity (rad/s) or Angular acceleration (rad/s^2) so the direction in Angular displacement (selected unit) stays correct.

Mistake: Reading only Angular displacement (degrees) after using the arc-length method and then plugging that number into another physics formula that expects radians.
Fix: Use Angular displacement (radians) for formulas like arc length and rotational kinematics, and check Notes (unit conversions used) if you are unsure.

• The arc-length method assumes motion along a circle and requires Radius (meters) to be greater than 0.
• The formula using Angular acceleration (rad/s^2) assumes constant angular acceleration during the whole time interval.
• "Wrapped smallest signed change" keeps the direction of the shortest rotation. "Wrapped smallest unsigned change (0 to 180 deg)" gives its magnitude. Only "Wrapped difference in chosen range" uses "Wrap range (chosen-range policy)".
• For the arc-length method, the calculator works in radians internally even if you choose degree output.
• Signed inputs are allowed. Negative Arc length traveled (meters), Angular velocity (rad/s), Initial angular velocity (rad/s), or Angular acceleration (rad/s^2) represent direction, so negative outputs are possible.
• Rounded display settings can hide tiny differences, especially for very small or very large values.
• The calculator does not infer real-world direction conventions from a diagram; it only follows the signs and units you enter.

Core formulas

The calculator uses one of four standard relationships based on your selected Calculation method.

Delta θ = θend - θstart

For the start/end method, raw change is end minus start and preserves multiple turns. Shortest-change policies return the signed shortest rotation or its unsigned magnitude. "Wrapped difference in chosen range" instead maps the difference into the selected full-turn interval.

θrad = s / r

For arc length and radius, angular displacement in radians equals arc length divided by radius. This relationship uses radians by definition, so the calculator finds the radian value first and only then converts to degrees if needed.

θrad = ω x t

For angular velocity and time, angular displacement equals angular velocity times elapsed time. Angular velocity is the rate of change of angle with time.[1]

θrad = ω0 x t + 0.5 x α x t^2

For constant angular acceleration, the calculator uses initial angular velocity, angular acceleration, and time to find the total angular displacement.

Unit conversions

rad = deg x (π/180)

deg = rad x (180/π)

rev = rad / (2π)

rad/s = deg/s x (π/180)

rad/s = rev/s x 2π

rad/s = rpm x 2π / 60

If you enter angular velocity or initial angular velocity in deg/s, rev/s, or rpm, the calculator converts them to rad/s before using the formulas. The result card then shows the conversion in Notes (unit conversions used).

Wrapped angle difference logic

For Angle difference policy (start/end method), raw keeps the exact value of end minus start. Wrapped smallest signed change returns the shortest signed rotation. Wrapped smallest unsigned change returns the size of that shortest rotation without a sign. This is especially useful for cases like 350 degrees to 10 degrees, where the raw difference is -340 degrees but the smallest signed change is +20 degrees.[2]

Mini example

Suppose you choose the arc-length method with Arc length traveled (meters) = 1.5 and Radius (meters) = 0.5.

θrad = 1.5 / 0.5 = 3

θdeg = 3 x (180/π) = 171.8873...

turns = 3 / (2π) = 0.4775...

So the outputs are 3 radians, about 171.9 degrees, and about 0.4775 revolutions.

Assumptions used in the calculator

The acceleration method assumes constant angular acceleration, the arc method assumes circular motion, and wrapped results depend on the policy you select. Hidden inputs are ignored for modes where they do not apply, and the arc method blocks invalid radius values such as 0 or negative numbers.[1]