Area of a sector and segment: Class 10 practice
A sector with angle θ is θ/360 of the circle: area = (θ/360)πr², arc = (θ/360)2πr. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Area of a circle
- Area of triangles
The idea behind area of a sector and segment
A sector with angle θ is θ/360 of the circle: area = (θ/360)πr², arc = (θ/360)2πr. A segment is the sector minus the triangle formed by the two radii.
Step-by-step method
- Find θ and r.
- Sector area = (θ/360)πr².
- For a segment subtract the triangle's area (½r² sin θ).
- Give units.
Worked example 1 (Basic)
Find the area of a sector of radius 28 cm and angle 60° (π = 22/7).
- (60/360) × 22/7 × 28²
- = 410.67 cm²
- Answer: 410.67 cm²
Worked example 2 (Standard)
Find the arc length of a sector of radius 21 cm and angle 120° (π = 22/7).
- (120/360) × 2 × 22/7 × 21
- = 44.00 cm
- Answer: 44.00 cm
Worked example 3 (Challenge)
Find the area of the minor segment cut by a chord subtending 90° in a circle of radius 28 cm (π = 22/7).
- Sector = 616.00 cm²
- Triangle = ½ × 28 × 28 = 392 cm²
- Segment = 224.00 cm²
- Answer: 224.00 cm²
Common mistakes and misconceptions
- Using 180 instead of 360
- Forgetting to subtract the triangle
Practice questions
- (Basic) Find the area of a sector of radius 21 cm and angle 45° (π = 22/7).
- (Basic) Find the area of a sector of radius 14 cm and angle 120° (π = 22/7).
- (Standard) Find the arc length of a sector of radius 21 cm and angle 60° (π = 22/7).
- (Standard) Find the arc length of a sector of radius 21 cm and angle 45° (π = 22/7).
- (Challenge) Find the area of the minor segment cut by a chord subtending 90° in a circle of radius 21 cm (π = 22/7).
- (Challenge) Find the area of the minor segment cut by a chord subtending 90° in a circle of radius 7 cm (π = 22/7).
Answer key with full working
- Q1: (45/360) × 22/7 × 21² → = 173.25 cm² → Answer: 173.25 cm²
- Q2: (120/360) × 22/7 × 14² → = 205.33 cm² → Answer: 205.33 cm²
- Q3: (60/360) × 2 × 22/7 × 21 → = 22.00 cm → Answer: 22.00 cm
- Q4: (45/360) × 2 × 22/7 × 21 → = 16.50 cm → Answer: 16.50 cm
- Q5: Sector = 346.50 cm² → Triangle = ½ × 21 × 21 = 220.5 cm² → Segment = 126.00 cm² → Answer: 126.00 cm²
- Q6: Sector = 38.50 cm² → Triangle = ½ × 7 × 7 = 24.5 cm² → Segment = 14.00 cm² → Answer: 14.00 cm²
Syllabus note
This skill is listed under "Areas Related to Circles" for Class 10 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is area of a sector and segment in?
"Areas Related to Circles", Class 10 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.