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5.3PureStretch

Areas of Sectors and Segments

In radians the area of a sector is a clean fraction of the whole circle. Subtracting the triangle from the sector gives the area of a segment — a common exam question.

30 min Radians
Edexcel A Level Maths: 5.3 Area of Sectors and SegmentsCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Use A = ½r²θ for sector area
  • Find the area of a segment
  • Combine with arc length results
  • Solve problems with unknown r or θ
1

Area of a sector

For radius rr and angle θ\theta in radians, the sector area is A=12r2θA = \tfrac12 r^2\theta. (It is the fraction θ2π\tfrac{\theta}{2\pi} of the full circle area πr2\pi r^2.)

A=12r2θ(θ in radians)A = \tfrac12 r^2\theta \quad (\theta \text{ in radians})
Sector area.
2

Area of a segment

A segment is the region between a chord and the arc. Its area is the sector minus the triangle formed by the two radii: A=12r2θ−12r2sin⁡θ=12r2(θ−sin⁡θ)A = \tfrac12 r^2\theta - \tfrac12 r^2\sin\theta = \tfrac12 r^2(\theta - \sin\theta).

Asegment=12r2(θ−sin⁡θ)A_{\text{segment}} = \tfrac12 r^2(\theta - \sin\theta)
Sector minus triangle.
Example 1Find the area of a segment with r=6r = 6 cm and θ=π3\theta = \tfrac{\pi}{3}.
112(6)2(π3−sin⁡π3)\tfrac12 (6)^2\left(\tfrac{\pi}{3} - \sin\tfrac{\pi}{3}\right).
2=18(1.047−0.866)≈18(0.181)= 18\left(1.047 - 0.866\right) \approx 18(0.181).
Answer
≈3.26\approx 3.26 cm²

Tip — Triangle area between two radii is ½r²sin θ (using two sides and the included angle).

Formula recap

A=12r2θA = \tfrac12 r^2\theta
Sector area.
A=12r2(θ−sin⁡θ)A = \tfrac12 r^2(\theta - \sin\theta)
Segment area.

Common mistakes to avoid

Using degrees in A = ½r²θ.
θ must be in radians.
Forgetting to subtract the triangle for a segment.
Segment = sector − ½r²sin θ.

Key takeaways

  • Sector area: A = ½r²θ (θ in radians).
  • Segment area: ½r²(θ − sin θ).
  • Triangle between two radii: ½r²sin θ.

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 area of a sector (θ in radians) is:

    • 12r2θ\tfrac12 r^2\theta
    • rθr\theta
    • πr2\pi r^2
    • 12rθ\tfrac12 r\theta
    Show answer and explanation

    Answer: 12r2θ\tfrac12 r^2\theta

    Standard formula.

  2. The area of a segment is the sector minus:

    • the triangle
    • the arc
    • a radius
    • the chord
    Show answer and explanation

    Answer: the triangle

    Segment = sector − triangle.

  3. The triangle between two radii has area:

    • 12r2sin⁡θ\tfrac12 r^2\sin\theta
    • 12r2θ\tfrac12 r^2\theta
    • r2sin⁡θr^2\sin\theta
    • 12rsin⁡θ\tfrac12 r\sin\theta
    Show answer and explanation

    Answer: 12r2sin⁡θ\tfrac12 r^2\sin\theta

    ½ab sin C with a=b=r.

  4. Segment area formula:

    • 12r2(θ−sin⁡θ)\tfrac12 r^2(\theta - \sin\theta)
    • 12r2(θ+sin⁡θ)\tfrac12 r^2(\theta + \sin\theta)
    • 12r2θ\tfrac12 r^2\theta
    • r2(θ−sin⁡θ)r^2(\theta - \sin\theta)
    Show answer and explanation

    Answer: 12r2(θ−sin⁡θ)\tfrac12 r^2(\theta - \sin\theta)

    Sector minus triangle.

  5. Sector r = 4, θ = π/2; area is:

    • 4π4\pi
    • 8π8\pi
    • 2π2\pi
    • π\pi
    Show answer and explanation

    Answer: 4π4\pi

    ½·16·π/2 = 4π.

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