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Area of Sectors

Maths: Edexcel A Level Pure Maths, topic 5.1.3

Area of a Sector of a Circle

As with the arc length, we can also use angles to find the area of a sector of a circle.

Derivation

Derivation

  • The area of a circle with radius rr is equal to πr2\pi r^2.
  • If the sector angle is θ\theta radians, then θ2π\frac{\theta}{2\pi} is the sector angle as a fraction of the angle in a full circle.
  • This means the area of the sector as a fraction of the area of the circle must also be equal to θ2π\frac{\theta}{2\pi}.
Derivation cont.

Derivation cont.

  • So the area of a sector is equal to:
  • Aπr2=θ2π→A=12r2θ\frac{\small A}{\small \pi r^2} = \frac{\small \theta}{\small 2\pi} \rightarrow A= \frac{1}{2}r^2\theta
Example

Example

  • What is the area of a sector of a circle with radius 6 cm and angle θ=π4\theta = \frac{\pi}{4} rad?
Area of sector

Area of sector

  • The area of a sector is A=12r2θA = \frac{1}{2}r^2\theta.
  • Substituting in, we have:
  • A=12×62×π4=4.5π=14.14A = \frac{1}{2}\times6^2\times\frac{\pi}{4} = 4.5\pi = 14.14 cm2

Test yourself on Area of Sectors

  • What is the equation for the area of a sector of a circle?

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