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Area of Segments

Maths: Edexcel A Level Pure Maths, topic 5.1.4

Area of a Segment of a Circle

We can use the area of a sector as well as the area of a triangle to find the area of a segment of a circle.

Derivation

Derivation

  • A sector of a circle of radius rr and angle θ\theta has points A and B on the circumference of the circle.
  • AB forms a segment of the circle.
  • The area of the segment is given by:
  • Area of segment = area of sector − area of triangle ABC.
Derivation cont.

Derivation cont.

  • The area of a triangle is equal to 12absin⁡C\frac{1}{2}ab\sin C, which for ABC, is equal to:
  • 12r2sin⁡θ\frac{1}{2}r^2\sin\theta
  • The area of the segment is then equal to:
  • Area of segment = 12r2θ−12r2sin⁡θ=12r2(θ−sin⁡θ)\frac{1}{2}r^2\theta - \frac{1}{2}r^2\sin\theta = \frac{1}{2}r^2(\theta - \sin\theta)
Example

Example

  • What is the area of the shaded segment formed by the line PQPQ?
Area of segment

Area of segment

  • The area of the segment is equal to 12r2(θ−sin⁡θ)\frac{1}{2}r^2(\theta - \sin\theta).
  • Substituting in, we have:
  • Area of segment = 12×42(π6−sin⁡π6)=12×16×0.0235...\frac{1}{2}\times4^2\left(\frac{\pi}{6}-\sin\frac{\pi}{6}\right) = \frac{1}{2}\times16\times0.0235...
  • Area of segment = 0.19 cm2

Test yourself on Area of Segments

  • What is the area of a segment formed from a sector of a circle with radius rr and an angle of θ\theta radians?

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