Wrong shortcode initialized

I. Chapter Summary:

This chapter explores formulas and applications related to the areas and perimeters of circular shapes. It introduces students to the concepts of circles, sectors, and segments and teaches them to calculate areas and perimeters of composite figures involving circles. The emphasis is on applying geometry and mensuration in practical, real-life contexts.

II. Key Concepts Covered:

Concept	Description
Circumference of a Circle	Total boundary length of a circle: $2pi r$
Area of a Circle	The space enclosed within a circle: $pi r^2$
Area of Sector	Portion of a circle enclosed by two radii and the arc: $left( frac{theta}{360} right) times pi r^2$
Length of an Arc	The curved portion of a circle’s boundary: $left( frac{theta}{360} right) times 2pi r$
Area of Segment	Region between a chord and the arc: Area of sector – Area of triangle
Composite Figures	Shapes formed by combining sectors, segments, circles, rectangles, etc.

III. Important Questions:

(A) Multiple Choice Questions (1 Mark):

1. What is the circumference of a circle with radius 7 cm?
   a) $14pi , text{cm}$
   b) $49pi , text{cm}$
   c) $28pi , text{cm}$
   d) $14pi , text{cm}$ ✅
2. The area of a circle of radius 14 cm is:
   a) 616 cm² ✅
   b) 88 cm²
   c) 154 cm²
   d) 44 cm²
   (PYQ 2019)
3. What is the area of a semicircle of radius r?
   a) $pi r^2$
   b) $frac{1}{2} pi r^2$✅
   c) $2pi r$
   d) $pi r$
4. The length of an arc of a sector of radius 10 cm and angle 90° is:
   a) $5pi , text{cm}$
   b) $10pi , text{cm}$
   c) $10pi , text{cm}$ ✅
   d) $2pi , text{cm}$

(B) Short Answer Questions (2/3 Marks):

1. Find the area of a sector of angle 60° and radius 7 cm.
   (PYQ 2020)
2. Find the length of an arc of a circle of radius 14 cm subtending an angle of 45° at the center.
3. A circle of radius 10.5 cm is cut into two semicircles. Find the perimeter of one semicircle.
4. A quadrant of a circle of radius 14 cm is cut out. Find its area.

(C) Long Answer Questions (5 Marks):

1. A horse is tied to a peg in the corner of a square field of side 15 m with a rope of length 5 m. Find the area the horse can graze.
   (PYQ 2022)
2. Find the area of the shaded region in a rectangle of size $10, text{cm} times 7, text{cm} = 70, text{cm}^2$
   with semicircular ends on both shorter sides.
3. In a circle of radius 10 cm, find the area of the major segment formed by a chord of length 10 cm.
4. A circular flower bed is surrounded by a path of uniform width 3.5 m. If the radius of the flower bed is 10.5 m, find the area of the path.

(D) HOTS (Higher Order Thinking Skills):

1. A circular pizza is cut into 8 equal sectors. One slice is removed. Calculate the area of the remaining pizza and its perimeter. What fraction of the original pizza is left?
2. Two roads, each 5 m wide, cross at right angles in a rectangular park 100 m × 60 m. Calculate the area of the remaining park excluding the roads.

IV. Key Formulas/Concepts:

Formula	Use
Circumference $= 2pi r$	Boundary length of a circle
Area of Circle $= pi r^2$	Area enclosed by circle
Area of Sector $= left( frac{theta}{360} right) times pi r^2$	Area covered by a sector
Length of Arc $= left( frac{theta}{360} right) times 2pi r$	Curved arc length of sector
Area of Segment = Area of Sector – Area of Triangle	For segment calculations
Use $pi approx 3.14 quad text{or} quad frac{22}{7}$	Based on radius divisibility

V. Deleted Portions (CBSE 2025–2026):

No portions have been deleted from this chapter as per the rationalized NCERT textbooks.

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–26):

Unit/Chapter	Estimated Marks	Type of Questions Typically Asked
Areas Related to Circles	4–6 Marks	1 Short Answer + 1 Long Answer or HOTS/Case-based

VII. Previous Year Questions (PYQs):

Year	Marks	Question
2019	1M	Area of circle from radius 14 cm
2020	3M	Area of sector with given radius and angle
2021	5M	Application-based grazing area question
2022	5M	Area of shaded region with circular segments

VIII. Real-World Application Examples:

Scenario	Concept Used
Designing circular fountains or gardens	Area of circle, sector
Roadside curves and roundabouts	Length of arc, segments
Cost estimation of circular tiles or circular carpets	Area calculations
Pizza or cake cutting	Sector and arc concepts

IX. Student Tips & Strategies for Success:

 Time Management:

• Spend 20 minutes on this chapter daily during revision weeks.
• Practice with real-life word problems.

 Exam Preparation:

• Memorize all formulas with units.
• Practice drawing figures correctly.
• Solve case-based and HOTS problems from past papers.

 Stress Management:

• Break down long problems into smaller steps.
• Use estimation to verify final answers.
• Revise formulas with flashcards or formula charts.

X. Career Guidance & Exploration (Class-Specific):

For Classes 9–10:

Stream	Example Careers	Entrance Exams
Science	Civil Engineer, Architect, Astronomer	NTSE, Olympiads
Commerce	Investment Analyst (geometry in stats, charts)	Olympiads
Arts	Interior Designer, Cartographer	Aptitude-based tests

Tip: Participate in Math Olympiads and ISRO/Science fairs where geometry applications are widely tested.

XI. Important Notes:

•  Always refer to the official CBSE and NCERT websites for updated syllabus.
•  Use diagrams and visualizations to understand composite figures.
•  Revise all formulas and solve 5–10 problems per week.
•  Conceptual clarity will help in solving application-based and case study questions.