IGCSE Mathematics: Circles, Arcs and Sectors Practice Questions
An arc is a fraction of the circumference and a sector is the same fraction of the area. That fraction is the sector angle divided by 360.
Topic 5.3 of Cambridge IGCSE Mathematics 0580 rests on one idea: work out what fraction of the circle you have, then apply it. The perimeter of a sector is where marks are lost, because the two radii must be added to the arc. The questions below cover that case explicitly.
What you need to know for Circles, Arcs and Sectors
- Circumference and areaCircumference = 2 pi r, or pi d. Area = pi r2. Check whether the question gives the radius or the diameter.
- Arc lengthArc = (angle divided by 360) x 2 pi r. It is the fraction of the circumference corresponding to the sector angle.
- Sector areaSector area = (angle divided by 360) x pi r2.
- Perimeter of a sectorArc length plus two radii. The two straight edges are part of the boundary.
- Working backwardsIf the arc length is given, substitute into the arc formula and solve for the angle or the radius.
- AccuracyUse the pi button rather than 3.14, keep full accuracy through the working and round only the final answer.
IGCSE Mathematics Circles, Arcs and Sectors questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
A circle has a radius of 7 cm. Calculate its circumference and its area, correct to 3 significant figures.
Show the worked answer
- Circumference = 2 pi r = 2 x pi x 7 = 43.98 cm, which is 44.0 cm to 3 significant figures.
- Area = pi r2 = pi x 72 = pi x 49.
- Area = 153.94 cm2.
- To 3 significant figures the area is 154 cm2. Note the different units: cm for circumference and cm squared for area.
A sector of a circle has a radius of 10 cm and an angle of 72 degrees at the centre. Calculate the arc length and the sector area, correct to 3 significant figures.
Show the worked answer
- The fraction of the circle is 72 divided by 360 = one fifth.
- Arc length = one fifth x 2 pi x 10 = one fifth x 62.83 = 12.566 cm, which is 12.6 cm.
- Sector area = one fifth x pi x 102 = one fifth x 314.16 = 62.83 cm2.
- To 3 significant figures the sector area is 62.8 cm2.
Calculate the perimeter of the same sector, radius 10 cm and angle 72 degrees, correct to 3 significant figures.
Show the worked answer
- The perimeter of a sector consists of the curved arc plus the two straight radii.
- The arc length is 12.566 cm from the previous calculation.
- The two radii contribute 10 + 10 = 20 cm.
- Perimeter = 12.566 + 20 = 32.566, which is 32.6 cm to 3 significant figures.
An arc of length 15 cm is drawn on a circle of radius 6 cm. Calculate the angle of the sector at the centre, correct to 1 decimal place.
Show the worked answer
- Write the arc formula: arc = (angle divided by 360) x 2 pi r.
- Substitute: 15 = (angle divided by 360) x 2 x pi x 6 = (angle divided by 360) x 37.699.
- Rearrange: angle divided by 360 = 15 divided by 37.699 = 0.39789.
- angle = 0.39789 x 360 = 143.24, which is 143.2 degrees to 1 decimal place.
Common mistakes in this topic
- Using the diameter in place of the radius, or the reverse.
- Omitting the two radii from the perimeter of a sector.
- Using the angle directly instead of as a fraction of 360.
- Giving an area answer in cm rather than cm squared.
- Rounding pi to 3.14 partway through, which shifts the final answer.
Exam tips
- Write the fraction angle over 360 as the first line of every sector calculation.
- Check whether the question gives radius or diameter before substituting.
- For a perimeter, trace the boundary of the sector: curve, straight, straight.
- Use the pi button on your calculator and keep full accuracy until the final answer.
- Sense check: an arc must be shorter than the full circumference, and a sector area smaller than the full circle.
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Circles, Arcs and Sectors FAQs
How do I calculate arc length?
Multiply the full circumference by the fraction of the circle the sector represents. That fraction is the sector angle divided by 360. So the arc length equals the angle over 360, multiplied by 2 pi r.
What is the perimeter of a sector?
The arc length plus the two radii. The sector is bounded by one curved edge and two straight edges, so all three must be added. Giving only the arc length is the most common error in this topic.
How do I find the sector angle from a given arc length?
Substitute the known arc length and radius into the arc formula, then rearrange to make the angle the subject. Divide the arc length by the full circumference to find the fraction of the circle, then multiply that fraction by 360.
Should I use 3.14 or the pi button?
Always use the pi button on your calculator and keep full accuracy through the working. Rounding pi to 3.14 partway through introduces error that can shift the third significant figure of your final answer and cost the accuracy mark.
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.