Textbook chapter 11
Calculate portions of circular regions and distinguish the curved boundary from a straight chord or radius.
Textbook exercise solutions
- Exercise 11.1 — 14 question groups
Videos, notes and tests
English · Session 2026–27 · NCERT-aligned study guide
Chapter overview
Calculate portions of circular regions and distinguish the curved boundary from a straight chord or radius.
Before you begin: Circle area and circumference, triangle area and degrees.
Key concepts · Solved examples · Practice · Mind map · Timed chapter test
Learn the key concepts
Sector and arc
A sector is enclosed by two radii and an arc. Its area and arc length take the fraction θ/360 of the full circle. A sector perimeter also includes both radii.
Segment
A minor segment lies between a chord and its minor arc. For the relevant central angles, find the sector area then subtract the triangle formed by the two radii and chord. At 90° that triangle is right-angled; at 60° it is equilateral.
Composite regions
Sketch the entire figure, label radii and angles, and state exactly which regions are added or removed. For a perimeter, trace only the exposed boundary. Keep π symbolic unless the question specifies a numerical value.
Formula and result checklist
- Sector area=(θ/360)πr².
- Arc length=(θ/360)2πr.
- Sector perimeter=2r+arc length.
- Minor segment area=sector area−area of the central triangle.
Solved examples
Example 1
Find the area and arc length of a 90° sector of radius 14 cm, using π=22/7.
Area=(1/4)(22/7)(196)=154 cm². Arc=(1/4)(2)(22/7)(14)=22 cm. Sector perimeter=22+28=50 cm.
Example 2
Find the minor segment area for radius 10 cm and central angle 90°.
Sector area=25π cm². The central right triangle has area (1/2)(10)(10)=50 cm². Segment area=25π−50 cm².
Example 3
Find the area of a ring with outer radius 7 cm and inner radius 4 cm.
Subtract disk areas: π(7²−4²)=33π cm². The ring has two circular boundaries if its full perimeter is needed.
Try it yourself
Work on paper before opening each answer. Show the reasoning and include units when needed.
1. Find area of a 60° sector of radius 6 cm.
Check answer 1
6π cm².
2. Find arc length for r=9 cm and θ=120°.
Check answer 2
6π cm.
3. Find the 60° minor segment area for r=6 cm.
Check answer 3
6π−9√3 cm²; subtract the equilateral triangle of side 6.
Common mistakes to avoid
- Arc length is measured in units; area in square units.
- A sector perimeter is not just its arc length.
Chapter mind map
View this chapter in the Mind Maps library · Open full-size image
Check your understanding
Take the 6-question, 10-minute chapter practice test. Each correct answer earns 1 mark; there is no negative marking. See explanations after submission.
Start chapter 11 practice test →
Continue learning
Read the corresponding chapter in the official NCERT textbook, then complete its exercises on paper. These NSL explanations and original practice questions support your textbook study; they are not an exercise-by-exercise solution manual.
Official CBSE syllabus 2026–27 · All chapters · Previous chapter · Next chapter
Check what you have learned
Finished the topics? Practise with the free chapter test.
Start free test