Circular Measure

Revise Circular Measure for Pure Mathematics 1 P1 (AS Level) — revision notes and instant AI marking.

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Cambridge International AS & A Level Mathematics 9709 — Pure Mathematics 1

Circular Measure

The big idea: Measuring angles in radians makes the arc length simply rθ and the sector area ½r²θ. Most questions then combine these with triangle facts to find perimeters and areas of shaded regions.

Summary — What This Chapter Covers

One radian is the angle subtended at the centre by an arc equal in length to the radius.
π radians = 180°.
Arc length s = rθ and sector area A = ½r²θ, with θ in radians.
Segment area = sector − triangle = ½r²(θ − sin θ).
Use trigonometry (sine rule, cosine rule, right-angled triangles) for chords and other lengths.

1. Radians, Arc Length and Sector Area

Radians

s = rθ and A = ½r²θ

Convert degrees to radians by multiplying by π/180 and radians to degrees by multiplying by 180/π. Learn the common ones: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 135° = 3π/4.

For a shaded segment (the region between a chord and an arc), subtract the area of triangle OAB = ½r² sin θ from the sector. The perimeter of a segment is the arc plus the chord; find the chord with the cosine rule or with 2r sin(θ/2).

Formulae (θ in radians) s = rθ · A = ½r²θ · triangle area = ½r² sin θ · chord = 2r sin(θ/2)

Worked Example

A sector has radius 10 cm and angle π/3 radians. Find the area and the perimeter of the segment cut off by the chord.

Practice Question 1

A sector of a circle of radius 5 cm has perimeter 20 cm. Find the angle of the sector and its area.

Practice Question 2

Express 135° in radians and 0.6 radians in degrees.

Calculator mode

Sine of an angle in radians needs radian mode. sin(π/3) in degree mode gives 0.0183 instead of 0.866, and the whole answer is lost.

Memorise

Conversion

Arc

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Circular Measure