Mensuration and calculation: calculate arc lengths, angles and areas of sectors of circles

The idea

A full circle has circumference 2πr and area πr². A sector is a fraction of the circle, determined by the angle θ at the centre. The arc length is (θ/360) × 2πr, and the sector area is (θ/360) × πr². You may also be asked to find the angle θ given the arc length or sector area, by rearranging these formulas. For example, a sector with radius 6 cm and angle 60° has arc length (60/360) × 2π × 6 = 2π cm and area (60/360) × π × 6² = 6π cm². Always use the radius, not the diameter, and ensure the angle is in degrees when using these fractions. When working with composite shapes, carefully identify which fraction of the circle is required and add any straight edges to find the total perimeter.

What earns the marks

• Uses the formula for arc length: arc length = (θ/360) × 2πr, where θ is the angle at the centre in degrees.
• Uses the formula for the area of a sector: area = (θ/360) × πr².
• Rearranges the arc length or sector area formula to find the angle θ when the arc length or area and radius are known.
• Applies these formulas to solve problems, including those involving composite shapes or perimeters of sectors.

Common mistakes and how to put them right

• Using the diameter instead of the radius in the formulas.

  Put it right: Check whether the given length is a radius or diameter; if it is a diameter, halve it before substituting.

• Forgetting to divide by 360 when finding a fraction of the circle.

  Put it right: The fraction of the circle is θ/360, so multiply the full circumference or area by this fraction.

• Mixing up the formulas for arc length and sector area.

  Put it right: Arc length uses 2πr (circumference) and area uses πr² (area of circle).

Exam tips

• Write down the formula you are using and substitute the values carefully, keeping the angle in degrees for the fraction θ/360.
• If you need to find the angle, rearrange the formula before substituting numbers to reduce arithmetic errors.
• Check that your answer is sensible: an arc length should be less than the full circumference, and a sector area should be less than the full circle area.