What is the formula for angular displacement?

The simplest formula is theta equals omega.*t, i.e. for uniform rotation you multiply the angular velocity by time. If the rotational speed changes with a constant angular acceleration then: theta = (omega-zero*t) + (1/2*alpha*t^2). Here omega-zero is the initial angular velocity and alpha is the angular acceleration. You can also calculate the angle from the arc length and radius using the formula theta = s / r. Alternatively you can find the angle by subtracting the final angle from the starting angle. This calculator uses whichever method fits best for your input values.

How big is the angular displacement if a wheel turns 3 revolutions?

A full rotation is equal to 360 degrees which is equivalent to 2 Pi radians. So the angular displacement for three rotations would be 3 multiplied by 360, which is 1080 degrees. This is equivalent to 6 Pi radians which is approximately 18.8 radians. If you enter 3 in the Angular Displacement field and set the unit to "Rotations", then the calculator will display 1080 degrees and 18.85 radians.

What is the difference between angular displacement and arc length?

The angular displacement is the rotated angle, measured in radians, and it's the same for every point of a rotating rigid body. The arc length is the actual distance that a particular point travels along a circular path, measured in meters, and it depends on how far this point is from the axis of rotation. Both quantities are related by the radius: the arc length is the product of angular displacement and radius, so s = θr.

Is the angular displacement a vector?

Yes. A full angular displacement has a magnitude (the rotated angle) and a direction (along the axis of rotation, determined by the right-hand rule). In most everyday cases it is enough to represent it with a sign (+ or -). A plus sign represents one sense of rotation, a minus sign the other. If the final angle is less than the starting angle, or if the motion reverses due to deceleration, this calculator will return a negative angle.

What unit is used to measure angular displacement?

The SI unit is the radian, with a full turn being 2π radians. In everyday use it can also be expressed in degrees, where a full turn is 360 degrees. It can also be expressed in turns, where one turn is equal to a full turn. One radian is approximately 57.3 degrees. This calculator will display the results simultaneously in radians, degrees and turns so you can read off the value in the unit you want.

Angular Displacement Calculator

Angular displacement

How do you want to work it out?

Pick what you measured. Only the fields that method needs are shown.

Angular velocity and time

Initial velocity, angular acceleration and time

Arc length and radius

Start angle and end angle

Angular velocity (ω)

Time (t)

Angular displacement (θ)

Enter an angular velocity and a time and the calculator finds the angle swept, θ = ω × t. Leave any one of the three blank and that is the one it works out.

Please input values into the fields

Angular displacement is the angle through which a rotating object has turned. When a wheel turns, when gears mesh or when a planet rotates, angular displacement is the angle that the object has rotated around its axis. This corresponds to the usual concept of linear displacement in rotational motion. While linear displacement measures how far an object has moved along a straight line, angular displacement measures how much an object has turned around an axis.

This calculator uses four methods to determine the angle and is suitable for any direction. If you leave one of the values blank, the calculator will calculate that value.

What is angular displacement?

Angular displacement is usually denoted by the Greek letter theta and it measures the change in angular position of a rotating object. If a point starts at one angle and ends up at another, then the angular displacement is the difference between those two angles.

In the International System of Units (SI), angular displacement is measured in radians but degrees and revolutions are also commonly used in everyday contexts. A radian is the angle subtended by an arc whose length is equal to the radius. Thus a full revolution is 2π radians, or 360 degrees or one turn. Angular displacement is actually a vector; its magnitude is the angle, its direction lies along the axis of rotation and is determined by the right-hand rule. In everyday applications it suffices to indicate the direction with a sign: A plus sign indicates one sense of rotation, a minus sign the other.

Formula for angular displacement.

There are various methods to calculate the change of angle and they are all integrated into this calculation tool. If the rotational speed is constant then the angle corresponds to the product of the angular velocity and time.

\theta = \omega\, t

Where omega is the angular velocity, or how fast an object is spinning, and t is time. If the rotational speed changes with a constant angular acceleration, then the second term is needed to account for the extra angle that comes from the accumulation of acceleration.

\theta = \omega_0\, t + \tfrac{1}{2}\,\alpha\, t^2

ω0 is the initial angular velocity, α is the angular acceleration and t is time. The first term represents the angle swept out by an object with constant angular velocity ω0 while the second term represents the additional rotation due to acceleration or deceleration.

The change of angle is also related to the usual length traveled by a point on a circular path. The arc length is the product of the angle expressed in radians and the radius, so that the angle is the result of dividing the arc length by the radius.

\theta = \dfrac{s}{r} \qquad s = \theta\, r

Since angular change is a change in angle position it can always be calculated by subtracting the final angle from the initial angle.

\theta = \theta_2 - \theta_1

Example calculation

A wheel starts at rest and rotates with a constant angular acceleration of 3.0 radians per second squared. After 8 seconds, what angle has it gone through?

We use the formula for acceleration. The initial angular velocity is zero, the angular acceleration is 3.0 and the time is 8 seconds.

\theta = 0\times 8 + \tfrac{1}{2}\times 3.0 \times 8^2 = 96 \ \text{rad}

So the wheel has rotated through 96 radians, which is about 15.3 revolutions. In the acceleration section you can enter initial angular velocity as 0, angular acceleration as 3 and time as 8 leaving angle blank. The calculator will return 96 radians, also showing the same result in degrees and revolutions.

How to use this calculator:

Select the method that matches your known information. For uniform rotation, enter the angular velocity and time in the first section to check the calculated angle.

For accelerated or decelerated rotation open the section for acceleration and enter initial angular velocity, angular acceleration and time. For rolling motion or circular motion when the distance traveled is known open the section for arc length and enter the arc length and radius. To find the angle between two positions open the last section and enter initial and final angles.

Each input field has a special option to convert units, so you can calculate any value by leaving the corresponding field blank. You can enter an angle in degrees and read out the time in seconds or enter an arc length in centimeters and read out the angle in radians. All conversions are done by a calculator tool.

Angle change units

The change of angle is measured as an angle and can be expressed in radians, degrees or revolutions. The radian is the SI unit and a unit required in physical formulas since it simplifies the relationship between arc length s = rθ.

Quantity	Symbol	SI unit
Angular displacement	θ	radian (rad); also degrees, revolutions
Angular velocity	ω	radian per second (rad/s); also rpm, rev/s
Angular acceleration	α	radian per second squared (rad/s²)
Time	t	second (s)
Arc length	s	metre (m)
Radius	r	metre (m)

Converting between units is easy. One complete revolution is equal to 2π radians, 360 degrees or one turn. For example if a wheel turns three times the angular displacement would be 3 * 360 = 1080 degrees. This is also equivalent to 6π radians which is approximately 18.8 radians. This calculator displays results in radians, degrees and turns side by side.

Distinguishing between angular change and linear displacement

It is helpful to distinguish between angular displacement and linear displacement. The angular displacement is the angle turned through and is measured in radians. Every point on a rigid body that is rotating has the same angular displacement. The wheel hub and the tire tread turn through the same angle.

The distance traveled along a curve, or arc length, is not the same at all points. Points farther from the axis describe a longer curve than points closer to the axis. This is because the arc length is the product of angle and radius. So for people who are farther from the center on a spinning carousel, it feels faster even though the entire structure goes through the same angle.

Where there are changes in direction

Whenever there is a rotation, an angular change takes place. The hands of the clock turn by a certain angle every hour, a drill rotates several times while drilling a hole and a satellite sweeps out a particular angle in its orbit.

Engineers use angular change to determine the size of gears or pulleys, calculate the rotational speed of a motor shaft, or determine distance traveled based on the number of revolutions of a robot wheel. In each case, information about the angle turned, either alone or in combination with radius, provides the necessary data.

Common Mistakes

A common mistake is to confuse radians and degrees. Since the formulas for angular velocity and acceleration use radians, angles given in degrees must be converted first. One radian is approximately equal to 57.3 degrees, not vice versa.

Another common mistake is to confuse angle and arc length. While the total angular displacement of an object is the same, the arc length depends on the radius. Also note that angular displacement can be negative. A negative value simply means that the object rotated in the opposite direction.

This tool is for general educational purposes and solving everyday problems. If used for research, engineering or safety related work please verify the values against the standards and data applicable to your project.

Frequently asked questions

What is the formula for angular displacement?: The simplest formula is theta equals omega.*t, i.e. for uniform rotation you multiply the angular velocity by time. If the rotational speed changes with a constant angular acceleration then: theta = (omega-zero*t) + (1/2*alpha*t^2). Here omega-zero is the initial angular velocity and alpha is the angular acceleration. You can also calculate the angle from the arc length and radius using the formula theta = s / r. Alternatively you can find the angle by subtracting the final angle from the starting angle. This calculator uses whichever method fits best for your input values.
How big is the angular displacement if a wheel turns 3 revolutions?: A full rotation is equal to 360 degrees which is equivalent to 2 Pi radians. So the angular displacement for three rotations would be 3 multiplied by 360, which is 1080 degrees. This is equivalent to 6 Pi radians which is approximately 18.8 radians. If you enter 3 in the Angular Displacement field and set the unit to "Rotations", then the calculator will display 1080 degrees and 18.85 radians.
What is the difference between angular displacement and arc length?: The angular displacement is the rotated angle, measured in radians, and it's the same for every point of a rotating rigid body. The arc length is the actual distance that a particular point travels along a circular path, measured in meters, and it depends on how far this point is from the axis of rotation. Both quantities are related by the radius: the arc length is the product of angular displacement and radius, so s = θr.
Is the angular displacement a vector?: Yes. A full angular displacement has a magnitude (the rotated angle) and a direction (along the axis of rotation, determined by the right-hand rule). In most everyday cases it is enough to represent it with a sign (+ or -). A plus sign represents one sense of rotation, a minus sign the other. If the final angle is less than the starting angle, or if the motion reverses due to deceleration, this calculator will return a negative angle.
What unit is used to measure angular displacement?: The SI unit is the radian, with a full turn being 2π radians. In everyday use it can also be expressed in degrees, where a full turn is 360 degrees. It can also be expressed in turns, where one turn is equal to a full turn. One radian is approximately 57.3 degrees. This calculator will display the results simultaneously in radians, degrees and turns so you can read off the value in the unit you want.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Angular displacement
Definition of angular displacement as the change in angular position, its vector nature, and its measurement in radians.
OpenStax University Physics, Rotation with Constant Angular Acceleration
The rotational kinematics equations, including the angle swept theta = omega-zero t + one half alpha t squared under a constant angular acceleration.
OpenStax University Physics, Angular Position and Displacement
Angular position, angular displacement and the arc-length relation s = theta r that connects the angle turned to the distance a point on the rim travels.