Area of a Sector

\( A = \frac{\theta}{360^{\circ}}\, \pi r^2 \)
Geometry GCSE
Question 6 of 20
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\( Radius 14 cm, angle 45^{\circ}. What is area of sector? \)

Hint (H)
\( Use formula A=(θ/360)×πr^2. \)

Explanation

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Statement

The formula for the area of a sector is:

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

Here, \(A\) is the area of the sector, \(\theta\) is the central angle in degrees, and \(r\) is the radius of the circle.

Why it’s true (short reason)

  • The full area of a circle is \(\pi r^2\).
  • A full turn is \(360^{\circ}\).
  • The sector with angle \(\theta\) represents a fraction \(\theta/360\) of the circle’s area.

Recipe (how to use it)

  1. Identify the radius \(r\) and the central angle \(\theta\).
  2. Write the formula \(A=\frac{\theta}{360}\pi r^2\).
  3. Substitute the known values.
  4. Calculate, keeping answers in terms of \(\pi\) if asked, or rounding if required.

Spotting it

  • Look for shaded “slice” or “sector” diagrams of a circle.
  • Phrases like “area of a sector”, “shaded area of circle”, or “fraction of circle” are indicators.
  • Sometimes the diameter is given — halve it to find the radius before using the formula.

Common pairings

  • Arc length: often appears alongside area of a sector in exam questions.
  • Perimeter of a sector: requires both arc length and two radii.
  • Fractions of a circle: the formula is essentially “fraction of circle area”.

Mini examples

  1. Given: \(r=7\), \(\theta=90^{\circ}\). Find: area of sector. Answer: \((90/360)\pi \times 49 = 49\pi/4.\)
  2. Given: \(r=3\), \(\theta=120^{\circ}\). Find: area. Answer: \((120/360)\pi \times 9 = 3\pi.\)

Pitfalls

  • Using diameter instead of radius: always check carefully.
  • Mixing arc length and area formulas: both use \(\theta/360\), but one multiplies by circumference, the other by area.
  • Radians vs degrees: at GCSE level, use degrees.
  • Rounding too early: keep \(\pi\) exact until the final step.

Exam strategy

  • Underline radius and angle values immediately in the question.
  • Check units: if radius is in cm, the area is in cm\(^2\).
  • If asked for perimeter of sector, don’t stop at area — add arc length and two radii.
  • Always state the formula before substituting — it secures method marks.

Summary

The area of a sector is found by taking the fraction of the circle’s area corresponding to the angle at the centre. Multiply that fraction by \(\pi r^2\). This formula is widely used for problems involving fractions of circles, shaded regions, and real-life applications like slices of pizza or wedges of a wheel.

Worked examples

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  1. \( Radius 7 cm, angle 90^{\circ}. Find area of sector. \)
    1. \( A=(90/360)×π×7^2. \)
    2. \( =(1/4)×49π. \)
    3. \( A=49π/4. \)
    Answer: \( \tfrac{49\pi}{4}\,\text{cm}^2 \)
  2. \( Radius 3 m, angle 120^{\circ}. Find area. \)
    1. \( A=(120/360)×π×3^2. \)
    2. \( =(1/3)×9π. \)
    3. \( A=3π. \)
    Answer: \( 3\pi\,\text{m}^2 \)
  3. \( Circle radius 10 cm, angle 60^{\circ}. Find area of sector. \)
    1. \( A=(60/360)×π×100. \)
    2. \( =(1/6)×100π. \)
    3. \( A=50π/3. \)
    Answer: \( \tfrac{50\pi}{3}\,\text{cm}^2 \)
  4. \( Radius 14 cm, angle 45^{\circ}. Find area. \)
    1. \( A=(45/360)×π×196. \)
    2. \( =(1/8)×196π. \)
    3. \( A=49π/2. \)
    Answer: \( \tfrac{49\pi}{2}\,\text{cm}^2 \)
  5. \( A circle has area 144π cm^2. Find area of a 90^{\circ} sector. \)
    1. \( Total area=144π. \)
    2. \( Sector=(90/360)×144π. \)
    3. \( =36π. \)
    Answer: \( 36\pi\,\text{cm}^2 \)
  6. \( Radius 5 cm, angle 150^{\circ}. Find sector area. \)
    1. \( A=(150/360)×π×25. \)
    2. \( =(5/12)×25π. \)
    3. \( A=125π/12. \)
    Answer: \( \tfrac{125\pi}{12}\,\text{cm}^2 \)
  7. \( Radius 6 cm, angle 225^{\circ}. Find area of sector. \)
    1. \( A=(225/360)×π×36. \)
    2. \( =(5/8)×36π. \)
    3. \( A=22.5π. \)
    Answer: \( 22.5\pi\,\text{cm}^2 \)
  8. \( Radius 40 cm, angle 18^{\circ}. Find area of sector. \)
    1. \( A=(18/360)×π×1600. \)
    2. \( =(1/20)×1600π. \)
    3. \( A=80π. \)
    Answer: \( 80\pi\,\text{cm}^2 \)
  9. \( Sector area=24π cm^2 in circle radius 12 cm. Find angle θ. \)
    1. \( A=(θ/360)×π×144. \)
    2. \( 24π=(θ/360)×144π. \)
    3. \( 24=(θ/360)×144 => θ=60. \)
    Answer: \( 60^{\circ} \)
  10. \( Sector area=64π cm^2 in circle radius 16 cm. Find angle θ. \)
    1. \( A=(θ/360)×π×256. \)
    2. \( 64π=(θ/360)×256π. \)
    3. \( 64=(θ/360)×256 => θ=90. \)
    Answer: \( 90^{\circ} \)