Arc Length of a Sector

\( L = \frac{\theta}{360^{\circ}}\, 2\pi r \)
Geometry GCSE
Question 2 of 20
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\( Radius 15 cm, angle 30^{\circ}. Find arc length. \)

Hint (H)
One-twelfth of circumference.

Explanation

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Statement

The formula for the arc length of a sector of a circle is

\[ L = \frac{\theta}{360^{\circ}} \times 2\pi r \]

Here, \(L\) is the arc length, \(\theta\) is the angle at the centre in degrees, and \(r\) is the radius of the circle.

Why it’s true (short reason)

  • The full circumference of a circle is \(2\pi r\).
  • A full turn around the centre measures \(360^{\circ}\).
  • If the angle is only \(\theta\), then the arc length is the same fraction of the circumference: \(\theta/360\) of \(2\pi r\).

Recipe (how to use it)

  1. Identify the radius \(r\) and the angle at the centre \(\theta\).
  2. Write the full circumference \(2\pi r\).
  3. Multiply it by the fraction \(\theta/360\).
  4. Simplify to find the arc length. Round to suitable decimal places if needed.

Spotting it

  • Look for questions about a sector, often shaded in a circle diagram.
  • Words such as “arc length” or “perimeter of a sector” are key triggers.
  • If only diameter is given, halve it to find the radius.

Common pairings

  • Sector area: similar formula, but with \(\pi r^2\) instead of \(2\pi r\).
  • Perimeter of a sector: add two radii to the arc length.
  • Radians: in higher work, the formula simplifies to \(L = r\theta\) when \(\theta\) is in radians.

Mini examples

  1. Given: \(r=7\), \(\theta=90^{\circ}\). Find: arc length. Answer: \((90/360)\times 2\pi \times 7 = 11\) (approx.).
  2. Given: \(r=3\), \(\theta=120^{\circ}\). Find: arc length. Answer: \((120/360)\times 2\pi \times 3 = 6.28\).

Pitfalls

  • Using diameter instead of radius: always check which is given.
  • Forgetting to divide by 360: without the fraction, the result is the whole circumference.
  • Mixing degrees and radians: at GCSE, the formula uses degrees.
  • Premature rounding: keep \(\pi\) exact until the last step.

Exam strategy

  • Underline the given angle and radius immediately in the question.
  • Check if they want the answer in terms of \(\pi\) or as a decimal.
  • If the question asks for “perimeter of the sector”, remember to add two radii.
  • Write the formula before substituting numbers — examiners often award a method mark just for that.

Summary

The arc length of a sector is found by taking the fraction of the circle corresponding to the given angle. Multiply that fraction by the circumference. Always check you are using the radius, divide the angle by 360, and keep results in exact form when requested.

Worked examples

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  1. \( Find the arc length of a sector with radius 7 cm and angle 90^{\circ}. \)
    1. \( L=(90/360)\times 2\pi \times 7. \)
    2. \( L=(1/4)\times 14\pi. \)
    3. \( L=3.5\pi. \)
    Answer: 3.5\pi cm
  2. \( Find the arc length of a sector with radius 3 m and angle 120^{\circ}. \)
    1. \( L=(120/360)\times 2\pi \times 3. \)
    2. \( =(1/3)\times 6\pi. \)
    3. \( L=2\pi. \)
    Answer: 2\pi m
  3. \( Radius 10 cm, angle 60^{\circ}. Find arc length. \)
    1. \( L=(60/360)\times 2\pi\times 10. \)
    2. \( = (1/6)\times 20\pi. \)
    3. \( L=10\pi/3. \)
    Answer: \tfrac{10\pi{3 cm
  4. \( Radius 14 cm, angle 45^{\circ}. Find arc length. \)
    1. \( L=(45/360)\times 2\pi\times 14. \)
    2. \( = (1/8)\times 28\pi. \)
    3. \( L=3.5\pi. \)
    Answer: 3.5\pi cm
  5. \( A circle has circumference 40\pi cm. Find arc length for angle 90^{\circ}. \)
    1. \( Full circumference=40\pi. \)
    2. \( Arc length=(90/360)\times 40\pi. \)
    3. \( =10\pi. \)
    Answer: 10\pi cm
  6. \( Find the arc length of a sector with r=5 cm, angle=150^{\circ}. \)
    1. \( L=(150/360)\times 2\pi\times 5. \)
    2. \( =(5/12)\times 10\pi. \)
    3. \( L=25\pi/6. \)
    Answer: \tfrac{25\pi{6 cm
  7. \( A circle radius 6 cm has a sector of 225^{\circ}. Find arc length. \)
    1. \( L=(225/360)\times 2\pi\times 6. \)
    2. \( =(5/8)\times 12\pi. \)
    3. \( L=7.5\pi. \)
    Answer: 7.5\pi cm
  8. \( A wheel of radius 40 cm rotates through angle 18^{\circ}. Find length of arc traced. \)
    1. \( L=(18/360)\times 2\pi\times 40. \)
    2. \( =(1/20)\times 80\pi. \)
    3. \( L=4\pi. \)
    Answer: 4\pi cm
  9. Radius 12 cm, arc length 8\pi cm. Find angle at centre.
    1. \( L=(θ/360)\times 2\pi r. \)
    2. \( 8\pi=(θ/360)\times 24\pi. \)
    3. \( 8=(θ/360)\times 24. \)
    4. \( θ=120. \)
    Answer: \( 120^{\circ} \)
  10. Arc length is 5\pi cm in circle radius 15 cm. Find angle θ.
    1. \( 5\pi=(θ/360)\times 30\pi. \)
    2. \( 5=(θ/360)\times 30. \)
    3. \( θ=60. \)
    Answer: \( 60^{\circ} \)