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Radians & Angular Displacement

Key idea: Express angular displacement in radians and use the arc-length relation s = rθ in A Level circular motion.

  • GCE A-Level H2 Physics 2027
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Learning objectives

  • Express angular displacement in radians and use s = rθ.

1. Definitions (Must Know)

A. Radian (rad)

One radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius.

So for an arc length s on a circle of radius r:

θ = s/r

Angles in circular motion formulas must be in radians.

B. Angular displacement, θ

Angular displacement, θ (rad), is the angle swept out at the centre of the circle.

Angular displacement measured by arc lengthA circle of radius r contains an arc of length s subtending an angle theta at the centre. A second highlighted arc has length r and therefore subtends exactly one radian.rarc sθdivide by rθ = s/rif s = rθ = 1 rad2π rad per turn
Scroll diagram horizontally to read all labels.
Radians connect rotation to distance directly: θ = s/r, so an arc equal to the radius subtends 1 rad.

2. Key Ideas (What Earns Marks)

  • Convert degrees to radians before using s = rθ.
  • One full circle is 2π rad, so:
    • 360° = 2π rad
    • 180° = π rad
  • The radian definition gives both θ = s/r and s = rθ.
  • State a rotational sign convention if direction matters; for example, anticlockwise positive.
Exam pitfall: degrees in radian formulas

Convert to radians before substituting into s = rθ. A calculator’s angle mode does not convert a number you have already entered as degrees.

3. Detailed Explanations

A. Why radians are “natural” for circles

Because θ = s/r, radians are a ratio of two lengths, so they are dimensionless.

That is why formulas like s = rθ only work cleanly when θ is in radians (not degrees).

B. Converting degrees to radians

Use:

θ (rad) = θ (°) × π/180

Mini-example:

30° = 30 × π/180 = π/6 rad

C. Signed angular displacement

Angular displacement describes the change in angular position. If a sign convention is needed, state it before calculating. For example, with anticlockwise positive, a quarter-turn clockwise is

Δθ = -π/2 rad.

The distance travelled along the arc is a scalar and remains positive:

s = r|Δθ|.

D. From arc length to angular speed

If an object moves an arc length s in time t:

  • θ = s/r
  • average angular speed ω = θ/t

For uniform motion, dividing s = rθ by the same time interval gives:

v = s/t = rθ/t = rω

The next lesson develops angular speed, period and frequency fully.

Arc length vs angle (radians)

A straight-line graph of arc length s against angle theta for a fixed radius. The gradient equals the radius.

Scroll across the graph to read all labels.

A straight-line graph of arc length s against angle theta for a fixed radius. The gradient equals the radius.A straight-line graph of arc length s against angle theta for a fixed radius. The gradient equals the radius.
Example with r = 0.50 m: s = rθ, so the graph is a straight line with gradient 0.50. This only works directly when θ is in radians.
Open full-size graph
View figure data
Values for Arc length vs angle (radians)
Angle, θ (rad)s = rθ (r = 0.50 m)
00
1.5710.785
3.1421.571
4.7122.356
6.2833.142

4. Common Mistakes

  • Using degrees inside s = rθ.
  • Using diameter where the formula requires radius.
  • Confusing angular displacement with the distance travelled around the arc.

5. Exam Tips

  • Keep exact multiples of π until the final step when possible.
  • State the positive rotational sense in questions where direction matters.
  • Check that θ = s/r is dimensionless; metres divide by metres.

6. Worked Examples

Modelled example 1

Convert degrees to radians

Core

Problem

Convert 135° to radians, retaining an exact multiple of π.
Study the worked solution

  1. Use the full-turn equivalence

    Method

    Multiply degrees by π/180.

    Reason

    180° = π rad.

    Working

    135° × π/180°

  2. Simplify

    Method

    Reduce 135/180 to 3/4.

    Reason

    The degree unit cancels and radians remain.

    Working

    θ = 3π/4 rad

Guided practice 2

Arc length from an angle

About 5 min

Problem

A point moves through 120° on a circle of radius 0.75 m. Find the arc distance.

Try this before viewing the solution

Unit: m

Hints

Hint 1: convert before substituting
120° = 2π/3 rad.
View solution step by step

  1. Convert angle

    Method

    Write 120° = 2π/3 radians.

    Reason

    The arc-length relation requires a radian angle.

    Working

    θ = 2π/3.

  2. Calculate arc length

    Method

    Use s = rθ.

    Reason

    A radian angle is the ratio s/r.

    Working

    s = (0.75)(2π/3) = 1.57 m

Common misconception 3

Degrees in s = rθ

Find and correct the mistake

Learner working

For r = 0.40 m and θ = 60°, a learner writes s = (0.40)(60) = 24 m. Locate the first error and correct the result.

Try this before viewing the solution

First error
Unit: m

View solution step by step

  1. Correct the angle unit

    Method

    Convert 60° to π/3 radians.

    Reason

    s = rθ is valid with θ in radians.

    Working

    θ = π/3.

  2. Recalculate

    Method

    Multiply 0.40 by π/3.

    Reason

    The radius is already in coherent SI units.

    Working

    s = 0.40(π/3) = 0.419 m

Examiner practice 4

Angular displacement with direction

3 marks

Examination question

Anticlockwise is positive. A wheel turns 1.75 revolutions clockwise. Find its angular displacement. [3 marks]

Try this before viewing the solution

View solution step by step

  1. Convert revolutions

    1 mark

    Method

    Use 2π radians per revolution.

    Reason

    A full turn subtends 2π radians.

    Working

    1.75(2π) = 3.5π.

  2. Assign direction

    1 mark

    Method

    Apply a negative sign.

    Reason

    Clockwise is opposite the stated positive sense.

    Working

    Δθ = -3.5π.

  3. State the quantity

    1 mark

    Method

    Include radians.

    Reason

    Angular displacement is signed and measured in radians.

    Working

    Δθ = -3.5π rad

Challenge 5

Angular displacement from arc length

Minimal support

Changed-unknown transfer

A point travels 1.20 m along a circle of radius 0.80 m. Find its angular displacement in radians and degrees.

Try this before viewing the solution

Hints

Hint 1: reverse the defining ratio
Rearrange s = rθ to θ = s/r, then convert radians to degrees.
View solution step by step

  1. Find the radian angle

    Method

    Divide arc length by radius.

    Reason

    A radian measure is defined by the ratio s/r.

    Working

    θ = 1.20/0.80 = 1.50 rad

  2. Convert representation

    Method

    Multiply by 180/π.

    Reason

    This converts radians to degrees.

    Working

    θ = 1.50180/π = 85.9°

7. Mind Stretchers

Mind stretcher 1: What goes wrong if you use degrees in s = rθ?Extension

Suppose an angle is 60°. A student substitutes θ = 60 into s = rθ instead of converting to radians.

By what factor is their arc length wrong?

Show Answer

The correct angle in radians is: θ = 60° × π/180 = π/3

Correct arc length is sₜᵣᵤₑ = r(π/3).

The student’s arc length is s_wrong = r(60).

So: s_wrong/sₜᵣᵤₑ = 60/(π/3) = 180/π ≈ 57.3

They overestimate the arc length by a factor of about 57.3.

Continue with the next resource in this course.

Course and syllabus information
Course
GCE A-Level H2 Physics
Edition
GCE A-Level H2 Physics 2027