CAIE A-Level · Mathematics 9709 · Circular Measure

Arc Length and Sector Area — Circular Measure (9709 Pure Mathematics 1)

9 min readSyllabus 1.4PreviewBy Uzair Khan

Syllabus objective

Use the formulae s = rθ and A = ½r²θ in solving problems concerning the arc length and sector area of a circle (including calculation of lengths and angles in triangles and areas of triangles).

Introduction

Arc length and sector area sit at the heart of the Circular Measure topic in 9709 Pure Mathematics 1. Once angles are expressed in radians (the prerequisite), two elegant formulae replace the more cumbersome degree-based versions. Examiners regularly combine these formulae with triangle geometry — finding perimeters of composite shapes, areas of segments, or lengths inside triangles — so fluency here unlocks marks across the whole topic.

Core Concept

Consider a circle of radius rr with a sector formed by two radii and the arc between them. The angle at the centre, measured in radians, is θ\theta.

  • The arc is the curved boundary of the sector.
  • The chord is the straight line joining the two endpoints of the arc.
  • The segment is the region between the chord and the arc.

Because θ\theta is in radians, the arc length is simply the fraction θ2π\dfrac{\theta}{2\pi} of the full circumference 2πr2\pi r, which simplifies to rθr\theta. Similarly, the sector area is θ2π\dfrac{\theta}{2\pi} of the full circle area πr2\pi r^2, giving 12r2θ\tfrac{1}{2}r^2\theta.

Critical requirement: θ\theta must always be in radians when using these formulae. Converting degrees to radians first is essential.

For the triangle formed by the two radii and the chord, you use standard triangle formulae. Since two sides are both equal to rr and the included angle is θ\theta:

Area of triangle=12r2sin⁡θ\text{Area of triangle} = \tfrac{1}{2}r^2\sin\theta

This leads directly to the segment area:

Area of segment=12r2θ−12r2sin⁡θ=12r2(θ−sin⁡θ)\text{Area of segment} = \tfrac{1}{2}r^2\theta - \tfrac{1}{2}r^2\sin\theta = \tfrac{1}{2}r^2(\theta - \sin\theta)

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Prerequisites: Radians and Degrees

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