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5.2PureCore

Arc Length

When the angle is measured in radians, the length of a circular arc is simply the radius times the angle. This clean formula is the main reason radians are preferred over degrees.

25 min Radians
Edexcel A Level Maths: 5.2 Arc LengthCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Use s = rθ for arc length
  • Work with θ in radians
  • Find perimeters of sectors
  • Solve for an unknown radius or angle
1

The arc length formula

For a sector of radius rr with angle θ\theta in radians\textbf{radians}, the arc length is s=rθs = r\theta. The angle must be in radians — convert first if given in degrees.

s=rθ(θ in radians)s = r\theta \quad (\theta \text{ in radians})
Arc length = radius × angle.
2

Perimeter of a sector

The perimeter of a sector is the arc plus the two radii: P=rθ+2rP = r\theta + 2r.

Example 1A sector has radius 55 cm and angle 0.80.8 rad. Find the arc length and perimeter.
1s=5×0.8=4s = 5 \times 0.8 = 4 cm.
2P=4+2(5)=14P = 4 + 2(5) = 14 cm.
Answer
Arc 44 cm, perimeter 1414 cm.

Tip — Perimeter includes both straight radii — don’t forget the +2r.

Formula recap

s=rθs = r\theta
Arc length (θ in radians).
P=rθ+2rP = r\theta + 2r
Sector perimeter.

Common mistakes to avoid

Using s = rθ with θ in degrees.
θ must be in radians for s = rθ.
Forgetting the two radii in a sector perimeter.
Perimeter = arc + 2r.

Key takeaways

  • s = rθ with θ in radians.
  • Convert degrees to radians first.
  • Sector perimeter = rθ + 2r.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. The arc length formula (θ in radians) is:

    • s=rθs = r\theta
    • s=12r2θs = \tfrac12 r^2\theta
    • s=2πrs = 2\pi r
    • s=r2θs = r^2\theta
    Show answer and explanation

    Answer: s=rθs = r\theta

    Radius × angle.

  2. Arc of a sector r = 5, θ = 0.8 rad:

    • 44
    • 88
    • 2.52.5
    • 4040
    Show answer and explanation

    Answer: 44

    5 × 0.8.

  3. For s = rθ, θ must be in:

    • radians
    • degrees
    • gradians
    • either
    Show answer and explanation

    Answer: radians

    Formula needs radians.

  4. Perimeter of a sector is:

    • rθ+2rr\theta + 2r
    • rθr\theta
    • 2r2r
    • rθ+rr\theta + r
    Show answer and explanation

    Answer: rθ+2rr\theta + 2r

    Arc plus two radii.

  5. If s = 12 and r = 4, then θ =

    • 33
    • 4848
    • 0.330.33
    • 88
    Show answer and explanation

    Answer: 33

    θ = s/r = 12/4.

Next step

Keep practising

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