Sheet № 224 · Higher only · AQA · Edexcel · OCR
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 an
§Key definitions
Question:
Find the area of a sector with radius 9 cm and angle 80°. Give your answer to 1 decimal place.
Answer:
The sector area is 56.5 cm².
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.
§Formulas to memorise
Area of a sector = (θ / 360) × πr²
Arc length = (θ / 360) × 2πr
Area of a segment = area of sector − area of triangle
Area of triangle in sector = ½r² sin θ
Substitute into: area = (θ / 360) × πr².
Subtract the triangle area from the sector area: segment = sector − triangle.
Worked example
Find the area of a sector with radius 9 cm and angle 80°. Give your answer to 1 decimal place.
This topic is Higher only, but this example uses a straightforward sector.
⚠ 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.