IGCSE Additional Mathematics: Circular Measure Practice Questions
In radians, arc length is r times theta and sector area is one half r squared theta. These forms are far simpler than the degree versions, which is why radians are used throughout Additional Mathematics.
Topic 10 of Cambridge IGCSE Additional Mathematics 0606 is high scoring and formula driven, provided the calculator is in radian mode. Segment area is the question that separates candidates. The questions below cover conversion, arc, sector and segment.
What you need to know for Circular Measure
- Radian conversion180 degrees equals pi radians. Multiply degrees by pi over 180 to convert, and by 180 over pi to reverse.
- Arc lengths = r theta, with theta in radians. No fraction of 360 is needed.
- Sector areaA = one half r2 theta, with theta in radians.
- Segment areaSector area minus triangle area: one half r2(theta minus sin theta), with theta in radians.
- Perimeter of a sectorArc length plus two radii, so r theta + 2r.
IGCSE Additional Mathematics Circular Measure questions and answers
4 exam-style questions written to the 0606 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Convert 135 degrees into radians, giving your answer as an exact multiple of pi.
Show the worked answer
- Multiply by pi over 180: 135 x pi divided by 180.
- Simplify the fraction 135 over 180 by dividing both by 45, giving 3 over 4.
- So 135 degrees equals 3 pi over 4 radians, approximately 2.356 radians.
A sector has radius 8 cm and angle 1.2 radians. Calculate the arc length and the sector area.
Show the worked answer
- Arc length: s = r theta = 8 x 1.2 = 9.6 cm.
- Sector area: A = one half r2 theta.
- A = 0.5 x 82 x 1.2 = 0.5 x 64 x 1.2.
- A = 38.4 cm2. Note that neither formula involves 360, because the angle is already in radians.
A sector has radius 10 cm and angle 1.5 radians. Calculate the area of the segment cut off by the chord, correct to 3 significant figures.
Show the worked answer
- The segment is the sector minus the triangle formed by the two radii and the chord.
- Segment area = one half r2(theta minus sin theta).
- = 0.5 x 100 x (1.5 minus sin 1.5). With the calculator in radian mode, sin 1.5 = 0.99749.
- = 50 x 0.50251 = 25.13, which is 25.1 cm2 to 3 significant figures.
Calculate the perimeter of a sector of radius 6 cm with angle 0.8 radians.
Show the worked answer
- The perimeter consists of the arc plus the two straight radii.
- Arc length = r theta = 6 x 0.8 = 4.8 cm.
- Two radii contribute 6 + 6 = 12 cm.
- Perimeter = 4.8 + 12 = 16.8 cm.
Common mistakes in this topic
- Leaving the calculator in degree mode.
- Using the 360 degree fraction formulas when the angle is in radians.
- Giving a decimal when an exact multiple of pi is requested.
- Omitting the two radii from a sector perimeter.
- Forgetting the one half in the sector area formula.
Exam tips
- Check the calculator mode before every circular measure question. It is the single biggest source of lost marks.
- Learn the three radian formulas. None of them involves 360.
- Segment equals sector minus triangle. Deriving it once makes the formula memorable.
- For exact answers, cancel fractions of pi rather than evaluating them.
- Sense check: an arc must be shorter than the full circumference and a segment smaller than its sector.
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Circular Measure FAQs
How do I convert between degrees and radians?
Since 180 degrees equals pi radians, multiply a value in degrees by pi over 180 to convert to radians, and multiply a value in radians by 180 over pi to convert back. Simplify the resulting fraction rather than evaluating it when an exact answer is wanted.
What are the arc length and sector area formulas in radians?
Arc length equals the radius multiplied by the angle in radians. Sector area equals one half multiplied by the radius squared multiplied by the angle in radians. Neither involves a fraction of 360, which is why radians simplify the working.
How do I find the area of a segment?
Subtract the area of the triangle formed by the two radii and the chord from the area of the sector. This gives one half r squared multiplied by theta minus sine theta, with the angle in radians and the calculator in radian mode.
Why does my segment answer come out wrong?
Almost always because the calculator is in degree mode when the sine is evaluated. The angle theta is used as a number in the bracket and as an angle inside the sine, so both must be in radians for the formula to work.
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Written to the published Cambridge IGCSE Additional Mathematics (0606) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.