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 for circumference and area of a circle, so if you are confident with those you are already halfway there. This page gives you the formulas, step-by-step methods, worked examples at both tiers, and plenty of practice. For a full list of formulas to learn, see our GCSE Maths formulas guide.

What Are Arc Length and Sector Area?

An arc is a portion of the circumference of a circle. The arc length is the distance along that curved portion.

A sector is the region enclosed by two radii and the arc between them. The sector area is the amount of space inside that region.

Both depend on two things: the radius of the circle and the angle at the centre of the sector.

Key Formulas

The fraction θ / 360 tells you what fraction of the full circle the sector represents. If the angle is 90°, the sector is 90/360 = 1/4 of the circle.

Perimeter of a Sector

Be careful — the perimeter of a sector is not just the arc length. It is:

Perimeter = arc length + 2 × radius

You add both straight edges (the two radii) as well as the curved edge.

Step-by-Step Method

Finding Arc Length

1. Write down the formula: arc length = (θ / 360) × 2πr.
2. Substitute the angle θ and the radius r.
3. Calculate the fraction first, then multiply by the circumference.
4. Round to the required degree of accuracy.

Finding Sector Area

1. Write down the formula: sector area = (θ / 360) × πr².
2. Substitute the angle θ and the radius r.
3. Calculate the fraction first, then multiply by the full area.
4. Round to the required degree of accuracy.

Working Backwards

Sometimes you are given the arc length or sector area and asked to find the angle or the radius. Rearrange the formula:

• To find θ: θ = (arc length / 2πr) × 360
• To find r from area: r² = (sector area × 360) / (θ × π)

Worked Example 1 — Foundation Level

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.

Working:

Arc length = (90 / 360) × 2 × π × 8 = (1/4) × 16π = 4π = 12.6 cm (1 d.p.)

Sector area = (90 / 360) × π × 8² = (1/4) × 64π = 16π = 50.3 cm² (1 d.p.)

Answer: Arc length = 12.6 cm, sector area = 50.3 cm².

Worked Example 2 — Higher Level

Question: A sector has an area of 40 cm² and a radius of 6 cm. Find the angle of the sector. Give your answer to the nearest degree.

Working:

Step 1 — Write the formula and substitute: 40 = (θ / 360) × π × 6² 40 = (θ / 360) × 36π

Step 2 — Rearrange to find θ: θ / 360 = 40 / (36π) θ = 360 × 40 / (36π) θ = 14400 / (36π) θ = 14400 / 113.097… θ = 127° (nearest degree)

Answer: The angle of the sector is 127°.

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.

Practice Questions

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.

Question 4: The arc length of a sector is 15 cm and the radius is 9 cm. Find the angle of the sector to the nearest degree.

Question 5: A sector has an angle of 150° and an area of 60 cm². Find the radius of the sector. Give your answer to 1 decimal place.

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Related Topics

• Circle Theorems — angle rules for circles (Higher only).
• Area and Perimeter — the foundation formulas for circles.
• Trigonometry in 3D — advanced geometry involving circles and angles.
• Constructions and Loci — drawing arcs with a compass.

Summary

Arc length and sector area calculations are based on finding a fraction of the full circle. The fraction is always θ / 360, where θ is the angle at the centre. Arc length uses the circumference formula and sector area uses the area formula, each multiplied by that fraction. Remember that the perimeter of a sector includes the two radii as well as the arc. These questions appear at both tiers and are typically worth 2–4 marks, so learning the formulas and practising rearrangements is essential for picking up straightforward marks in the exam.