Angular Displacement Calculator – Calculate Angular Displacement Formula

  • Adam
  • November 27, 2025

Calculate angular displacement instantly using arc length, angular velocity, or angular acceleration. Free calculator with step-by-step solutions and formulas. Convert between radians, degrees, and revolutions.

Angular Displacement Calculator - Calculate Angular Displacement Formula

Angular displacement is the angle through which an object rotates around a fixed point or axis, measured in radians, degrees, or revolutions. This calculator helps you determine angular displacement using various parameters including arc length, radius, angular velocity, angular acceleration, and time, making it essential for analyzing rotational motion in physics and engineering.

Calculate Angular Displacement

Result:

What is Angular Displacement?

Angular displacement is a vector quantity that describes the change in angular position of a rotating object, representing the angle through which a point, line, or body has rotated in a specified direction about a specified axis. Unlike linear displacement which measures straight-line distance, angular displacement measures rotational change and is typically expressed in radians, though degrees and revolutions are also common units.

Key Characteristics of Angular Displacement:

  • Vector quantity: Has both magnitude and direction (clockwise or counterclockwise)
  • Path-independent: Only the initial and final positions matter, not the path taken
  • Measured in radians: SI unit is the radian (rad), a dimensionless unit where 2π radians = 360 degrees = 1 complete revolution
  • Can be positive or negative: Convention typically treats counterclockwise rotation as positive and clockwise as negative
  • Analogous to linear displacement: Angular displacement is to rotation what linear displacement is to straight-line motion

Angular Displacement Formula

Angular displacement can be calculated using different formulas depending on which parameters are known. The fundamental relationships connect angular displacement to arc length, radius, angular velocity, angular acceleration, and time.

Basic Angular Displacement Formula

The most fundamental formula relates angular displacement to the arc length traveled and the radius of the circular path:

Angular Displacement from Arc Length:

θ = s / r

Where:

  • θ = angular displacement (radians)
  • s = arc length or distance traveled along the circular path (m)
  • r = radius of the circular path (m)

Angular Displacement from Angular Velocity

When angular velocity is constant, angular displacement is simply the product of angular velocity and time:

θ = ωt

Where:

  • θ = angular displacement (rad)
  • ω = angular velocity (rad/s)
  • t = time (s)

Angular Displacement with Angular Acceleration

When angular acceleration is involved, the kinematic equations for rotational motion apply:

First Equation (with initial velocity and acceleration):

θ = ω₀t + ½αt²

Second Equation (without time):

θ = (ωf² - ωi²) / 2α

Third Equation (average velocity):

θ = ½(ωi + ωf)t

Where:

  • θ = angular displacement (rad)
  • ω₀ or ωi = initial angular velocity (rad/s)
  • ωf = final angular velocity (rad/s)
  • α = angular acceleration (rad/s²)
  • t = time (s)

Angular Displacement from Positions

Δθ = θf - θi

Where:

  • Δθ = angular displacement (rad)
  • θf = final angular position (rad)
  • θi = initial angular position (rad)

Conversion Formulas

Revolutions to Radians:

θ (rad) = n × 2π

Where n is the number of revolutions

Degrees to Radians:

θ (rad) = θ (deg) × π / 180

Radians to Degrees:

θ (deg) = θ (rad) × 180 / π

How to Calculate Angular Displacement

Calculating angular displacement requires identifying which parameters are known and selecting the appropriate formula. Follow these systematic steps:

  1. Identify known quantities: Determine which values you have—arc length, radius, angular velocity, angular acceleration, time, or angular positions.
  2. Select the appropriate formula: Choose the angular displacement equation that matches your known variables.
  3. Ensure consistent units: Convert all measurements to SI units (radians for angles, meters for length, seconds for time).
  4. Substitute values: Plug the known values into the selected formula, maintaining proper signs for direction.
  5. Calculate the result: Perform the mathematical operations to find angular displacement.
  6. Verify direction and magnitude: Check that the sign indicates the correct direction of rotation (positive for counterclockwise, negative for clockwise by convention).
  7. Convert units if needed: Express the result in the desired unit system (radians, degrees, or revolutions).

Calculation Example 1: From Arc Length

Problem: A point on a wheel travels 12 meters along the circular rim. If the wheel has a radius of 0.5 meters, what is the angular displacement?

Solution:

Given: s = 12 m, r = 0.5 m

Using formula: θ = s / r

θ = 12 / 0.5

θ = 24 radians

Converting to degrees: θ = 24 × (180/π) = 1375.1 degrees

Converting to revolutions: θ = 24 / (2π) = 3.82 revolutions

Calculation Example 2: From Angular Velocity

Problem: A motor shaft rotates at a constant angular velocity of 20 rad/s for 5 seconds. Calculate the angular displacement.

Solution:

Given: ω = 20 rad/s, t = 5 s

Using formula: θ = ωt

θ = 20 × 5

θ = 100 radians

This equals approximately 15.92 revolutions or 5729.6 degrees.

Calculation Example 3: With Angular Acceleration

Problem: A flywheel starts from rest and accelerates at 2 rad/s² for 10 seconds. Find the angular displacement.

Solution:

Given: ω₀ = 0 rad/s (starts from rest), α = 2 rad/s², t = 10 s

Using formula: θ = ω₀t + ½αt²

θ = 0(10) + ½(2)(10)²

θ = 0 + ½(2)(100)

θ = 0 + 100

θ = 100 radians

Angular Displacement vs Angular Distance

Understanding the distinction between angular displacement and angular distance is crucial for correctly analyzing rotational motion:

Characteristic Angular Displacement Angular Distance
Type Vector quantity (has direction) Scalar quantity (magnitude only)
Definition Shortest angular path between initial and final positions Total angle covered regardless of direction
Path Dependence Path-independent (only endpoints matter) Path-dependent (total rotation counts)
Value Range Can be positive or negative Always positive or zero
Example 1.5 rotations clockwise = -3π rad 1.5 rotations = 3π rad

Applications of Angular Displacement

Angular displacement calculations are essential across numerous fields of science, engineering, and technology:

  • Mechanical Engineering: Analyzing gear rotations, shaft displacements, cam mechanisms, and rotary motion systems in machinery design
  • Robotics: Calculating joint rotations, servo motor positions, and robotic arm movements with precision control
  • Aerospace: Determining aircraft roll, pitch, and yaw angles, satellite orientation changes, and spacecraft attitude control
  • Automotive: Computing wheel rotations, steering angles, crankshaft positions, and engine timing in vehicle systems
  • Physics Education: Teaching rotational kinematics, circular motion, and the relationship between linear and angular quantities
  • Sports Biomechanics: Analyzing joint angles during movement, golf swing mechanics, and gymnastic rotations
  • Astronomy: Measuring celestial object positions, planetary rotations, and orbital angular displacements
  • Manufacturing: Controlling CNC machines, robotic welding arms, and automated assembly line rotational movements

Units of Angular Displacement

Angular displacement can be expressed in three primary unit systems, each suited to different applications:

Radians (rad):

The SI unit and most fundamental measure. One radian is the angle subtended at the center of a circle by an arc equal in length to the radius. A complete circle is 2π radians (approximately 6.28319 rad). Radians are dimensionless and preferred in mathematical and physical calculations because they simplify formulas.

Degrees (°):

More intuitive for everyday use, where a complete circle is 360 degrees. Commonly used in navigation, surveying, and engineering drawings. To convert: 1 radian = 180/π degrees ≈ 57.2958 degrees.

Revolutions (rev):

Counts complete rotations, useful for describing motor speeds and mechanical rotations. One revolution equals 2π radians or 360 degrees. Common in engineering specifications like RPM (revolutions per minute).

Relationship to Linear Displacement

Angular displacement is intimately connected to linear displacement for objects moving along circular paths. The arc length (linear distance) traveled by a point at distance r from the rotation axis relates to angular displacement through:

s = rθ

Linear displacement = radius × angular displacement (when θ is in radians)

This fundamental relationship means that points farther from the axis of rotation travel greater linear distances for the same angular displacement. For example, a point on the rim of a wheel travels farther than a point near the hub during one complete rotation, though both experience the same angular displacement of 2π radians.

Frequently Asked Questions

What is the difference between angular displacement and angular velocity?
Angular displacement is the total angle rotated (measured in radians, degrees, or revolutions), while angular velocity is the rate of change of angular position (measured in rad/s, deg/s, or RPM). Angular displacement tells you how far something has rotated, whereas angular velocity tells you how fast it is rotating. They are related by θ = ωt for constant angular velocity.
Can angular displacement be negative?
Yes, angular displacement can be negative. The sign indicates the direction of rotation. By standard convention, counterclockwise rotation is positive and clockwise rotation is negative (though this convention can vary depending on the coordinate system). A negative angular displacement simply means rotation occurred in the opposite direction to the chosen positive direction.
Why do we use radians instead of degrees for angular displacement?
Radians are preferred in physics and engineering because they are dimensionless and create simpler mathematical relationships. The formula s = rθ only works directly when θ is in radians, without needing conversion factors. Additionally, calculus operations involving trigonometric and rotational functions are simpler with radians. Radians are the natural unit derived from the ratio of arc length to radius.
How do you calculate angular displacement from RPM?
First convert RPM (revolutions per minute) to angular velocity in rad/s by multiplying by 2π/60. Then multiply by time in seconds: θ = (RPM × 2π/60) × t. For example, if a motor runs at 300 RPM for 10 seconds: ω = 300 × 2π/60 = 31.42 rad/s, then θ = 31.42 × 10 = 314.2 radians or 50 revolutions.
What is the angular displacement for one complete revolution?
One complete revolution equals 2π radians (approximately 6.283 radians), 360 degrees, or 1 revolution. These are equivalent expressions of the same angular displacement. For multiple revolutions, multiply by the number: n revolutions = 2πn radians = 360n degrees.
How is angular displacement related to arc length?
Angular displacement is related to arc length through the formula θ = s/r, where θ is angular displacement in radians, s is the arc length traveled, and r is the radius. This means angular displacement equals the arc length divided by the radius. Conversely, arc length equals radius times angular displacement (s = rθ) when using radians.
Can angular displacement be greater than 2π radians?
Yes, angular displacement can exceed 2π radians (360 degrees or one revolution). This simply means the object has completed more than one full rotation. For example, θ = 10π radians represents 5 complete revolutions. Unlike angular position which is often expressed modulo 2π, angular displacement accumulates with each rotation.
What is the formula for angular displacement with constant acceleration?
For constant angular acceleration, use θ = ω₀t + ½αt², where ω₀ is initial angular velocity, α is angular acceleration, and t is time. Alternatively, when time is unknown, use θ = (ωf² - ωi²)/(2α), where ωf is final angular velocity and ωi is initial angular velocity. These are the rotational equivalents of linear kinematic equations.
How do you find angular displacement from angular acceleration?
To find angular displacement from angular acceleration, you need additional information. If you know initial angular velocity (ω₀) and time (t), use θ = ω₀t + ½αt². If you know initial and final angular velocities (ωi and ωf) but not time, use θ = (ωf² - ωi²)/(2α). Without these additional parameters, angular acceleration alone cannot determine displacement.
What is the SI unit of angular displacement?
The SI unit of angular displacement is the radian (rad). Although technically dimensionless (since it's defined as a ratio of two lengths), the radian is treated as a derived SI unit for clarity. Other common units include degrees (°) and revolutions (rev), with conversions: 1 revolution = 2π radians = 360 degrees.

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