IGCSE Maths / Circles, cylinders, cones and spheres

Arc length of a sector

Edexcel International GCSE Maths (4MA1), chapter 12: Circles, cylinders, cones and spheres, specification point 4.9a. Free lesson.

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An arc is the curved edge of a sector, and its length is the same fraction of the circle’s circumference as the sector’s angle is of 360°. Choosing the circumference rather than the area formula, and giving the answer in centimetres, helps you avoid a common exam error.

What an arc length measures

A sector is the slice of a circle between two radii. Its curved edge is an arc. The angle at the centre is written as θ. Degrees are used for these questions. An arc is a length, so give its answer in cm, not cm².

How to find the arc length

The angle θ tells you what fraction of a full circle the sector covers. The arc is that same fraction of the circle’s circumference, which is 2πr. This formula is not printed on the formula sheet, so you need to know it. Arc length = (θ/360) × 2πr

How arc length differs from sector area

A sector’s angle gives the same fraction for its arc and its area, but the fraction is applied to a different circle measurement. Use 2πr for an arc, because it is a length. Use πr² for the area, because it covers a surface. For a radius of 12 cm and an angle of 45°, the fraction is 45/360 = 1/8. Arc length: 1/8 × 2π × 12 = 9.42 cm. Sector area: 1/8 × π × 12² = 56.5 cm².

How to work accurately with the fraction

Simplifying the angle fraction first makes the calculation easier. Keep enough figures during the calculation, then round the final answer to the requested accuracy.

What happens when the angle doubles

The angle θ appears on its own in the numerator of θ/360. If the angle doubles, the fraction doubles, so the arc length doubles too, as long as the radius stays the same.

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