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Class 10 Mathematics -> Chapter 11: Areas Related to Circles
I. Chapter Summary:
This chapter focuses on the area of a circle and the areas of sectors and segments of a circle. It introduces the formula for calculating the area of a circle, the area of a sector, and the area of a segment, along with the necessary application of these formulas to solve problems. The chapter also includes the concept of the perimeter of a sector and practical problems related to areas of circles. By the end of the chapter, students will be able to calculate the area and perimeter of sectors and segments, as well as apply these concepts in real-world contexts.
II. Key Concepts Covered:
- Area of a Circle:
- The area of a circle is given by the formula:
$text{Area} = pi r^2$
where r is the radius of the circle.
- Sector of a Circle:
- A sector is a region enclosed by two radii and the arc of a circle.
- The area of a sector of a circle with angle θthetaθ (in degrees) is given by:
Area of $text{Area of sector} = frac{theta}{360} times pi r^2$
- The perimeter of a sector is the sum of the two radii and the length of the arc:
Perimeter of $text{Perimeter of sector} = 2r + frac{theta}{360} times 2pi r$
- Segment of a Circle:
- A segment of a circle is the region bounded by a chord and the corresponding arc.
- The area of a segment is found by subtracting the area of the triangular portion from the area of the sector:
Area of $text{Area of segment} = text{Area of sector} – text{Area of triangle}$
- Application of Circle Formulas:
- Students will use these formulas to solve real-world problems, such as calculating the area of land or materials needed for circular fields or circular parts in construction.
III. Important Questions:
(A) Multiple Choice Questions (MCQs) (1 Mark):
- The area of a circle with radius 7 cm is:
- (A) $49pi , text{cm}^2$
- (B) $14pi , text{cm}^2$
- (C) $49 , text{cm}^2$
- (D) $7pi , text{cm}^2$
- Answer: (A)
- The length of the arc of a sector with radius 10 cm and central angle 60° is:
- (A) $10pi , text{cm}$
- (B) $10 , text{cm}$
- (C) $frac{10pi}{3} , text{cm}$
- (D) 5 cm
- Answer: (C)
- The area of a sector with radius 5 cm and central angle 90° is:
- (A) $frac{25pi}{2} , text{cm}^2$
- (B) $frac{25pi}{4} , text{cm}^2$
- (C) $25pi , text{cm}^2$
- (D) $frac{25pi}{3} , text{cm}^2$
- Answer: (B)
- The area of a segment of a circle is:
- (A) $text{Area of sector} – text{Area of triangle}$
- (B) Area of triangle−Area of sector
- (C) $pi r^2$
- (D) $pi r^2 – text{Area of chord}$
- Answer: (A)
(B) Short Answer Questions (2/3 Marks):
- Find the area of a sector of a circle with radius 7 cm and angle 60°.
- A sector of a circle has a radius of 10 cm and a central angle of 45°. Find the perimeter of the sector.
- Calculate the area of a segment of a circle with radius 8 cm and angle 90° if the area of the sector is $32pi , text{cm}^2$.
- The radius of a circle is 14 cm. Find the area of a sector with an angle of 120°.
(C) Long Answer Questions (5 Marks):
- A circular garden has a radius of 12 m. Find the area of a sector of the garden with a central angle of 45°. Also, find the perimeter of the sector.
- A circular wheel has a radius of 15 cm. A sector of the wheel subtends an angle of 60°. Calculate the area of the sector and the length of the arc.
- The radius of a circle is 10 cm. Find the area of the segment formed by a chord that subtends a central angle of 60°.
- A ring is formed by two concentric circles. The radius of the outer circle is 14 cm, and the radius of the inner circle is 7 cm. Find the area of the ring.
(D) HOTS (Higher Order Thinking Skills) Questions:
- A circle has a radius of 20 cm. A chord of length 24 cm is drawn. Find the area of the segment formed by the chord.
- A garden in the shape of a sector has a radius of 10 m and a central angle of 120°. If the cost of planting grass is Rs. 5 per square meter, calculate the cost of planting grass in the sector of the garden.
IV. Key Formulas/Concepts:
-
Area of a Circle:
Area of $text{Area of circle} = pi r^2$ -
Area of a Sector:
Area of $text{Area of sector} = frac{theta}{360} times pi r^2$Where $theta$ is the central angle in degrees.
-
Perimeter of a Sector:
Perimeter of $text{Perimeter of sector} = 2r + frac{theta}{360} times 2pi r$ -
Area of a Segment:
$text{Area of segment} = text{Area of sector} – text{Area of triangle}$ -
Length of Arc:
Length of $text{Length of arc} = frac{theta}{360} times 2pi r$
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 2025-2026):
| Unit/Chapter | Estimated Marks | Type of Questions Typically Asked |
|---|---|---|
| Areas Related to Circles | 6 | 1 Mark (MCQs), 2/3 Marks (Short Answers), 5 Marks (Long Answers) |
VII. Previous Year Questions (PYQs):
- 2019:
- (1 Mark) Find the area of a sector of a circle with radius 14 cm and central angle 60°.
- (2 Marks) A sector of a circle with radius 10 cm and central angle 90° has an area of $25π25pi25π cm²$
- Find the length of the arc of the sector.
- (5 Marks) A circular park has a radius of 21 m. Calculate the area of a sector of the park with a central angle of 90°. Also, find the perimeter of the sector.
- 2020:
- (3 Marks) The length of a chord of a circle is 12 cm, and the perpendicular distance from the center to the chord is 5 cm. Find the radius of the circle.
- (5 Marks) A circular garden has a radius of 15 m. Find the area of a sector with an angle of 120°. Also, calculate the perimeter of the sector.
VIII. Real-World Application Examples to Connect with Topics:
- Construction:
- In construction, trigonometric calculations are used to design circular parts of buildings, bridges, and domes, ensuring accurate dimensions and material use.
- Landscaping and Gardening:
- Landscapers use the formulas for areas of sectors and segments to calculate the amount of material, such as soil or grass, required to cover circular areas.
- Circular Roads and Tracks:
- Trigonometry helps in calculating the distances and areas for circular tracks, roads, and racetracks, especially when dealing with sectors of circles.
IX. Student Tips & Strategies for Success:
- Time Management:
- Focus on mastering the basic formulas first and apply them to various problems. Solve problems related to sectors and segments of circles regularly to reinforce your understanding.
- Exam Preparation:
- Practice word problems on areas of sectors, segments, and their applications. Work on deriving formulas for areas and perimeters for better retention.
- Stress Management:
- Break down complex problems into smaller parts. Stay calm during exams by practicing relaxation techniques like deep breathing.
X. Career Guidance & Exploration (Class-Specific):
- For Classes 9-10:
- The concepts learned in this chapter are foundational for students aspiring to pursue careers in Engineering, Architecture, and Mathematics. These topics are relevant for entrance exams such as JEE and NEET.
- For Classes 11-12:
- Students aiming for Engineering, Physics, Astronomy, and Architecture will find these concepts crucial. They will also be essential for preparing for JEE and other competitive exams.
XI. Important Notes:
- Consistent practice of different types of problems involving areas and perimeters is key to mastering this chapter.
- Refer to the official CBSE website for updated exam information and guidelines.
- Ensure you understand the derivation of formulas and their applications in real-world problems.