Sheet № 53 · Foundation + Higher · AQA · Edexcel · OCR
Arc Length and Sector Area –
Arc length and sector area questions appear on both Foundation and Higher tier GCSE Maths papers across AQA, Edexcel, and OCR. A sector is a "slice" of a circle (like a slice of pizza), and you need to be able to calculate the length of its curved edge (the arc) and the area it encloses. These calculations build directly on the formulas f
§Key definitions
Question:
A sector has a radius of 8 cm and an angle of 90°. Calculate the arc length and the area of the sector. Give your answers to 1 decimal place.
Answer:
Arc length = 12.6 cm, sector area = 50.3 cm².
Question 1:
A sector has a radius of 10 cm and an angle of 72°. Find the arc length. Give your answer to 1 decimal place.
Question 2:
A sector has a radius of 5 cm and an angle of 120°. Find the area of the sector. Give your answer to 1 decimal place.
Question 3:
Find the perimeter of a sector with radius 7 cm and angle 60°. Give your answer to 1 decimal place.
§Formulas to memorise
Arc length = (θ / 360) × 2πr, where θ is the angle in degrees and r is the radius
Sector area = (θ / 360) × πr², where θ is the angle in degrees and r is the radius
Circumference of a full circle = 2πr
Area of a full circle = πr²
Write down the formula: arc length = (θ / 360) × 2πr.
Write down the formula: sector area = (θ / 360) × πr².
To find θ: θ = (arc length / 2πr) × 360
To find r from area: r² = (sector area × 360) / (θ × π)
Worked example
A sector has a radius of 8 cm and an angle of 90°. Calculate the arc length and the area of the sector. Give your answers to 1 decimal place.
Working:
⚠ Common mistakes
- ✗Forgetting to use the fraction θ / 360. Students sometimes calculate the full circumference or full area instead of the fraction.
- ✗Confusing arc length with sector perimeter. The perimeter of a sector includes the two radii as well as the arc. If the question asks for the perimeter, remember to add 2r.
- ✗Using the diameter instead of the radius. The formulas use r (radius), not d (diameter). If you are given the diameter, halve it first.
- ✗Rounding π too early. Use the π button on your calculator rather than 3.14, which causes rounding errors.
- ✗Not giving units. Arc length is in cm (or m, mm, etc.) and area is in cm² (or m², mm², etc.). Always include the correct unit.
✦ Exam tips
- →Write the formula first — this earns a method mark even if you make an arithmetic slip later.
- →Show the fraction θ / 360 clearly in your working. Examiners look for this.
- →Sector perimeter questions catch many students out. Read the question carefully — "perimeter" and "arc length" are different things.
- →Leave your answer in terms of π if asked. Some questions say "Give your answer as a multiple of π." In that case, do not use a decimal.
- →For compound shapes, you may need to subtract one sector from another or combine a sector with a triangle. Break the shape into parts.