Area of a Sector and Segment –

Area of a sector and segment is a Higher-tier GCSE Maths topic tested on AQA, Edexcel, and OCR papers. Sector area uses a fraction of the full circle area, while segment area requires subtracting a triangle from a sector. This guide explains both calculations step by step, connects to the ½ab sin C formula, and provides worked examples and practice questions.

What Are Sectors and Segments?

• A sector is the region enclosed by two radii and an arc — like a pizza slice.
• A segment is the region between a chord and the arc it cuts off.

A minor sector or segment is the smaller one (angle less than 180°); a major sector or segment is the larger one.

Key Formulas

The triangle formed inside a sector has two sides equal to the radius (r) with the included angle θ, so you use Area = ½ × r × r × sin θ = ½r² sin θ.

Step-by-Step Method

Finding the Area of a Sector

1. Identify the radius (r) and the angle at the centre (θ).
2. Substitute into: area = (θ / 360) × πr².
3. Evaluate and round as required.

Finding the Area of a Segment

1. Calculate the area of the sector using (θ / 360) × πr².
2. Calculate the area of the triangle using ½r² sin θ.
3. Subtract the triangle area from the sector area: segment = sector − triangle.

Worked Example 1 — Foundation Level

This topic is Higher only, but this example uses a straightforward sector.

Question: Find the area of a sector with radius 9 cm and angle 80°. Give your answer to 1 decimal place.

Working:

Step 1 — Area = (80/360) × π × 9².

Step 2 — Area = (80/360) × 81π = (2/9) × 81π = 18π.

Step 3 — 18π ≈ 56.5 cm².

Answer: The sector area is 56.5 cm².

Worked Example 2 — Higher Level

Question: Find the area of the minor segment of a circle with radius 10 cm and central angle 120°. Give your answer to 1 decimal place.

Working:

Step 1 — Sector area = (120/360) × π × 10² = (1/3) × 100π = (100π/3) ≈ 104.720 cm².

Step 2 — Triangle area = ½ × 10² × sin 120° = 50 × sin 120° = 50 × (√3/2) = 25√3 ≈ 43.301 cm².

Step 3 — Segment area = 104.720 − 43.301 = 61.4 cm² (1 d.p.).

Answer: The segment area is 61.4 cm².

Worked Example 3 — Exam Style

Question: A circle has radius 8 cm. A chord AB subtends an angle of 150° at the centre O. Find the area of the major segment (the larger segment). Give your answer to 1 decimal place.

Working:

Step 1 — The minor sector angle is 150°. The minor sector area = (150/360) × π × 64 = (5/12) × 64π = (320π/12) = (80π/3) ≈ 83.776 cm².

Step 2 — The triangle area = ½ × 8² × sin 150° = 32 × 0.5 = 16 cm².

Step 3 — Minor segment area = 83.776 − 16 = 67.776 cm².

Step 4 — Full circle area = π × 64 = 64π ≈ 201.062 cm².

Step 5 — Major segment area = full circle − minor segment = 201.062 − 67.776 = 133.3 cm² (1 d.p.).

Answer: The major segment area is 133.3 cm².

Common Mistakes

• Forgetting to subtract the triangle for segment area. A segment is not the same as a sector. You must subtract the triangle: segment = sector − triangle.
• Using the wrong angle for the triangle. The triangle inside the sector uses the same angle θ as the sector — it is the angle between the two radii.
• Mixing up minor and major. If the question asks for the major segment, you need to subtract the minor segment from the full circle area.

Exam Tips

• The sector area formula (θ/360) × πr² is on the formula sheet, but ½r² sin θ for the triangle may not be — know how to derive it from ½ab sin C with a = b = r.
• For exact answers, leave in terms of π and √3 (for common angles like 60° and 120°).
• If the question gives the arc length instead of the angle, find θ first using arc length = (θ/360) × 2πr, then proceed.
• Always check whether the question asks for the minor or major segment/sector.

Practice Questions

Q1 (Higher): Find the area of a sector with radius 12 cm and angle 45°. Give your answer in terms of π.

Q2 (Higher): Find the area of the minor segment of a circle with radius 6 cm and angle 90°. Give your answer to 1 d.p.

Q3 (Higher): A sector has area 75π cm² and radius 15 cm. Find the angle of the sector.

Practise sector and segment area questions with instant feedback free on GCSEMathsAI.

Related Topics

• Arc Length and Sector Area — arc length calculations and sector area basics.
• Area Using ½ab sin C — the triangle area formula used for segments.
• Parts of a Circle — definitions of sector, segment, arc, and chord.

Summary

Sector area uses the fraction (θ/360) of the full circle area πr². Segment area requires an extra step: subtract the triangle formed by the two radii and chord from the sector area. The triangle area is ½r² sin θ, derived from the ½ab sin C formula with both sides equal to the radius. Always check whether the question asks for minor or major, and whether exact or decimal answers are required. Combining these formulas confidently is essential for Higher-tier success.