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06062025–2027 syllabus

ADDITIONAL MATHEMATICS · CHAPTER 9

Circular measure

Use radians as a natural ratio so arc, sector and segment calculations become direct.

Additional Mathematics4 connected sectionsSyllabus-aligned guide

LEARNING OBJECTIVES

What you will be able to do

  • convert between degrees and radians
  • calculate arc lengths and sector areas
  • find segment measures using triangles
  • solve compound circular-shape problems

AT A GLANCE

Syllabus0606Coverage2025–2027Sections4LevelAdditional Mathematics

INTRODUCTION · THE BIG IDEA

Use radians as a natural ratio so arc, sector and segment calculations become direct.

One radian is the angle subtended when the arc length equals the radius. Because the angle is defined as a ratio, radian formulas do not need a separate fraction of 360.

Circular-measure questions often combine sectors with triangles, chords or several radii. Mark every boundary before deciding which lengths and areas to add or subtract.

01

SECTION 01

Radians and conversions

Core concept

A full turn is 2π radians and a half-turn is π radians. Keep exact multiples of π whenever an exact angle is useful.

RULE 1
radians = degrees × π/180
RULE 2
degrees = radians × 180/π
RULE 3
θ = s/r
Original worked example

Convert and use a radian angle

  1. Convert 135° to radians.
  2. Multiply by π/180: 135π/180.
  3. Simplify the fraction by 45.

Answer: 135° = 3π/4 radians.

02

SECTION 02

Arc length and sector area

Core concept

For θ in radians, an arc has length rθ and a sector has area ½r²θ. Major sectors use 2π − θ when θ is the minor angle.

RULE 1
arc length s = rθ
RULE 2
sector area A = ½r²θ
RULE 3
sector perimeter = 2r + rθ
Sector with radius, central angle and arc labelledθrarcarc = θ/360 × 2πrsector = θ/360 × πr²sector perimeter = arc + 2r
A sector takes the same fraction θ/360 of both the full circumference and full circle area.
03

SECTION 03

Segments and chords

Core concept

A minor segment is the minor sector minus the isosceles triangle formed by two radii and the chord. The triangle area is ½r²sin θ.

DETAILED EXPLANATION

  • A major segment is the whole circle minus the minor segment.
  • The triangle formula uses the included central angle.
  • Check whether the requested perimeter includes radii.
RULE 1
minor segment area = ½r²(θ − sin θ)
RULE 2
chord length = 2r sin(θ/2)
RULE 3
minor segment perimeter = rθ + chord
04

SECTION 04

Compound circular regions

Core concept

Break a compound shape into named sectors, triangles and rectangles. Circle theorems from the assumed IGCSE Mathematics knowledge may supply missing angles before radian calculations begin.

ORIGINAL STUDY DIAGRAMBuild a compound answer
1Find all central angles
2Convert required angles to radians
3List boundary arcs and lines
4Add or subtract simple areas
5Attach correct units
Useful prerequisite geometry
FactUse
radius ⟂ tangentfind right angles
angle at centre = twice angle at circumferencerecover a central angle
angles in the same segment are equaltransfer an inscribed angle
angle in a semicircle is 90°form right triangles

QUICK CHAPTER SUMMARY

The ideas to carry forward

  • π radians equals 180°.
  • The formulas s = rθ and A = ½r²θ use radians.
  • Segment area is sector area minus triangle area.
  • Major and minor regions use complementary central angles.
  • Compound problems are solved by naming and combining simpler regions.

QUICK REVISION CHECKLIST

Can you do each of these without your notes?

  • convert between degrees and radians
  • calculate arc lengths and sector areas
  • find segment measures using triangles
  • solve compound circular-shape problems
← Coordinate geometry of the circleTrigonometry →