Circular measure

AS Level Mathematics · Pure Mathematics 1. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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The idea

A radian is another unit for angles. One radian is the angle at the centre of a circle when the arc is the same length as the radius. A full turn is 2π radians, so π radians is the same as 180°.

Radians make the formulae for a sector very short. The arc length and the sector area each need only the radius and the angle, but the angle must be in radians.

Harder questions join a sector to a triangle. A segment is the sector with the triangle taken away. A perimeter can include two radii and an arc, or a chord and an arc, so read the shape carefully.

Rules to remember

Common mistake

Students leave the calculator in degree mode when they work out sin θ for an angle in radians, or they put an angle in degrees into s = rθ. Check the mode before every calculation.

Worked example

A chord of a circle of radius 8 cm subtends an angle of 1.5 radians at the centre. Find the area of the minor segment, to 3 significant figures.

  1. Sector area = ½ × 82 × 1.5 = 48 cm2.
  2. Triangle area = ½ × 82 × sin 1.5 = 32 × 0.9975 = 31.92 cm2 (calculator in radian mode).
  3. Segment = 48 − 31.92 = 16.08 cm2.

Answer: 16.1 cm2

Practice questions

Try each one, then open the answer.

1. A sector of a circle of radius 8 cm has angle 1.2 radians. Find the area of the sector.

  1. A
    38.4 cm2
  2. B
    76.8 cm2
  3. C
    9.6 cm2
  4. D
    4.8 cm2
Show answer

Answer: A. Area = (1/2) r2 θ = (1/2) × 64 × 1.2 = 38.4 cm2. Leaving out the 1/2 gives 76.8.

2. A sector has radius 5 cm and angle 1.6 radians. Find the perimeter of the sector.

  1. A
    8 cm
  2. B
    13 cm
  3. C
    18 cm
  4. D
    26 cm
Show answer

Answer: C. Perimeter = two radii + arc = 2 × 5 + 5 × 1.6 = 10 + 8 = 18 cm. The common slip is forgetting one or both radii.

3. Convert 150° to radians, giving the answer in terms of π.

  1. A
    5π/12
  2. B
    2π/3
  3. C
    5π/6
  4. D
    3π/4
Show answer

Answer: C. 180° = π rad, so 150° = 150/180 × π = 5π/6.

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