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CIE A-Level Maths Study Notes

1.4.2 Arc Length and Sector Area

Arc Length

Arc length in a circle can be calculated using the formula s=rθs = r\theta, where ss represents the arc length, rr is the radius of the circle, and θ\theta is the central angle in radians.

arc length

Image courtesy to Cuemath

Example 1:

Calculate the arc length subtended by an angle of π4\frac{\pi}{4}radians in a circle with a radius of 4 units.

Solution:

Arc length =4×π4=π= 4 \times \frac{\pi}{4} = \pi units.

Example 2:

Determine the arc length when the central angle is π2\frac{\pi}{2}radians and the radius is 3 units.

Solution:

Arc length =3×π2=3π2= 3 \times \frac{\pi}{2} = \frac{3\pi}{2}units.

Area of a Sector

The area of a sector is given by A=12r2θA = \frac{1}{2} r^2 \theta, where AA is the area, rr is the radius, and θ\theta is the central angle in radians.

sector area

Image courtesy to Cuemath

Example 1:

Calculate the area of a sector where r=5r = 5 units and θ=π3\theta = \frac{\pi}{3} radians.

Solution:

Area =12×52×π3=25π6= \frac{1}{2} \times 5^2 \times \frac{\pi}{3} = \frac{25\pi}{6} square units.

Example 2:

Find the area of a sector with a radius of 7 units and a central angle of 2π3\frac{2\pi}{3} radians.

Solution:

Area =12×72×2π3=49π3= \frac{1}{2} \times 7^2 \times \frac{2\pi}{3} = \frac{49\pi}{3} square units.

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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