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MathidiaMathematics Academy
Circles and constructions
Lesson 52 of 75
Easy practice

Arcs and sectors

Objective. Calculate fractions of a circle's circumference and area.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

A sector is a slice of a circle bounded by two radii and an arc, and its arc length or area is found as a fraction of the whole circle's circumference or area, based on the angle at the centre. The fraction used is the sector's angle divided by 360°, since a full circle has 360° at its centre. For example, a sector with a 90° angle is a quarter of the circle, so its area is (90/360) × πr² and its arc length is (90/360) × 2πr. Correctly identifying the central angle given in the diagram is essential before applying the fraction to either formula.

Worked example

Find the arc length of a 120° sector of a circle with radius 9 cm. Give your answer in terms of π.

Diagram of a circle
  1. 1Arc length = (angle/360) × 2πr.
  2. 2The sector is 120/360 = 1/3 of the circle.
  3. 3Full circumference = 2π × 9 = 18π cm.
  4. 4Arc length = 1/3 × 18π = 6π cm.

Answer: 6π cm

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Common mistake. Pupils sometimes use the sector's angle divided by 180° instead of 360°, treating a full turn as a straight line rather than a full circle. Always divide the central angle by 360° for both arc length and sector area.

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