RD Sharma Solutions Class 9 Maths Chapter 16 – Download PDF
RD Sharma Solutions Class 9 Maths Chapter 16 - Circles are available in PDF format to help students better understand the concepts. A circle, in simple terms, is a closed curve on a plane, consisting of all points that are equidistant from a fixed point, known as the center. The center of the circle lies inside the circle. The distance between any point on the circumference and the center is called the radius. The diameter is the longest chord of the circle, which passes through the center.
RD Sharma Solutions provide students with the opportunity to practice problems covering key concepts such as the radius, segment, sector, and diameter of a circle.
Access the detailed RD Sharma Solutions for Class 9 Maths Chapter 16 and improve your chances of scoring high marks in your exams. The solutions have been carefully prepared by expert teachers according to the latest CBSE syllabus for the 2025-26 academic year.
RD Sharma Solution for Class 9 Chapter 16 - Download PDF
RD Sharma Solutions Class 9 Maths Chapter 16 Circles Solutions, designed to help students prepare effectively for their exams. By referring to these solutions and practicing the problems, students can boost their confidence and improve their scores.
RD Sharma Solution for Class 9 Chapter 16 - Question with Answers
1. What is a circle?
Answer: A circle is a closed plane curve where all points are equidistant from a fixed point, known as the center of the circle.
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2. What is the center of a circle?
Answer: The center of a circle is the fixed point that is equidistant from all points on the circumference of the circle.
3. What is the radius of a circle?
Answer: The radius of a circle is the distance between the center and any point on the circumference.
4. What is the diameter of a circle?
Answer: The diameter of a circle is the longest chord, which passes through the center and connects two points on the circumference. The diameter is twice the radius.
5. What is a chord in a circle?
Answer: A chord is a line segment with both endpoints on the circumference of the circle.
6. What is a sector of a circle?
Answer: A sector of a circle is the region enclosed by two radii and the arc between them.
7. What is a segment of a circle?
Answer: A segment of a circle is the region enclosed by a chord and the arc of the circle.
8. What is the arc of a circle?
Answer: An arc is a part of the circumference of a circle, defined by two points on the circle.
9. What is the formula for the circumference of a circle?
Answer: The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle.
10. What is the area of a circle?
Answer: The area of a circle is given by the formula A = πr², where r is the radius of the circle.
11. What is the relationship between the radius and diameter of a circle?
Answer: The diameter is twice the length of the radius, i.e., d = 2r.
12. How do you calculate the length of an arc?
Answer: The length of an arc is calculated using the formula Length of arc = (θ/360) × 2πr, where θ is the angle subtended by the arc at the center, and r is the radius.
13. How do you calculate the area of a sector?
Answer: The area of a sector is given by the formula Area of sector = (θ/360) × πr², where θ is the central angle in degrees, and r is the radius of the circle.
14. How is the area of a segment calculated?
Answer: The area of a segment is calculated by subtracting the area of the triangular portion from the area of the sector:
Area of segment = Area of sector - Area of triangle.
15. What is the relation between the angle subtended by a chord and the radius of the circle?
Answer: The angle subtended by a chord at the center of the circle is directly related to the size of the chord and the radius. Larger angles are subtended by longer chords.
16. How do you find the area of a circular ring?
Answer: The area of a circular ring is calculated by subtracting the area of the smaller circle from the area of the larger circle:
Area of ring = π(R² - r²), where R is the radius of the larger circle and r is the radius of the smaller circle.
17. What is a tangent to a circle?
Answer: A tangent is a straight line that touches the circle at exactly one point, known as the point of tangency.
18. What is the relationship between a tangent and the radius of the circle?
Answer: A tangent to a circle is always perpendicular to the radius at the point of contact.
19. What is the length of a tangent from a point outside the circle?
Answer: The length of the tangent from a point outside the circle can be found using the formula:
Length of tangent = √(OP² - r²), where O is the center of the circle, P is the external point, and r is the radius of the circle.
20. What is the angle subtended by a diameter of a circle at the circumference?
Answer: The angle subtended by the diameter of a circle at the circumference is always 90 degrees.
21. What is a cyclic quadrilateral?
Answer: A cyclic quadrilateral is a quadrilateral where all four vertices lie on the circumference of a circle.
22. What is the relationship between the angles of a cyclic quadrilateral?
Answer: In a cyclic quadrilateral, the sum of the opposite angles is always 180 degrees.
23. How do you calculate the area of a circular sector with a given radius and central angle?
Answer: The area of the sector is given by Area = (θ / 360) × πr², where θ is the central angle in degrees, and r is the radius.
24. What is the formula for the circumference of a semicircle?
Answer: The circumference of a semicircle is calculated as πr + 2r, where r is the radius.
25. What is the area of a semicircle?
Answer: The area of a semicircle is given by Area = 1/2 × πr², where r is the radius.
26. How do you calculate the area of a sector if the central angle is in radians?
Answer: The area of a sector with a central angle θ (in radians) is given by Area = 1/2 × r² × θ, where r is the radius.
27. What is the formula to find the area of a circle when the diameter is given?
Answer: The area of a circle is given by the formula Area = πd² / 4, where d is the diameter of the circle.
28. How do you calculate the area of a triangle inscribed in a circle?
Answer: The area of a triangle inscribed in a circle can be found using Heron’s formula or by using the relation Area = 1/2 × base × height, where the base is one side of the triangle, and the height is the perpendicular distance from the opposite vertex.
29. What is the formula for the circumference of a circle when the diameter is known?
Answer: The circumference of a circle can be calculated using the formula C = πd, where d is the diameter of the circle.
30. What is the relationship between a chord and a secant of a circle?
Answer: A chord is a line segment with both endpoints on the circumference, whereas a secant is a line that intersects the circle at two points.
31. How do you calculate the area of a sector with a central angle of 90 degrees?
Answer: For a 90-degree sector, the area is one-fourth of the area of the full circle:
Area = (1/4) × πr², where r is the radius.
32. What is the power of a point in relation to a circle?
Answer: The power of a point is the square of the length of the tangent from the point to the circle, and it is equal to the product of the distances from the point to the two intersection points on the circle.
33. What is the angle formed by two tangents drawn from an external point?
Answer: The angle between the two tangents drawn from an external point to the circle is always half of the angle subtended by the line joining the external point to the center of the circle.
34. What is the relationship between the radius and the length of a chord in a circle?
Answer: The radius of the circle is always longer than the length of any chord except the diameter.
35. How do you calculate the area of a sector of a circle when the central angle is given in degrees?
Answer: The area of the sector is Area = (θ / 360) × πr², where θ is the central angle in degrees and r is the radius.
36. What is the area of a segment of a circle?
Answer: The area of a segment of a circle is the area of the sector minus the area of the triangle formed by the two radii and the chord.
37. How do you find the length of the arc of a circle?
Answer: The length of the arc is given by the formula Length of arc = (θ / 360) × 2πr, where θ is the central angle in degrees and r is the radius.
38. What is the angle subtended by a chord at the center of the circle?
Answer: The angle subtended by a chord at the center of the circle is twice the angle subtended by the same chord at any point on the circumference.
Importance of RD Sharma Solutions for Class 9 Chapter 16 - Circles
RD Sharma Solutions for Class 9 Chapter 16 - Circles is an essential resource for students learning geometry. This chapter focuses on the concepts related to circles, such as the center, radius, diameter, chords, sectors, and arcs. Mastering these concepts is crucial for students to excel in both their exams and real-life applications. Here's why this chapter is important:
1. Conceptual Understanding of Circles: This chapter helps students build a solid foundation in geometry by introducing the properties and characteristics of a circle. Understanding the basic concepts like the center, radius, and diameter is key to solving more complex problems in higher grades.
2. Exam Preparation: The chapter Circles is a vital part of the Class 9 Maths syllabus, and questions based on circles are frequently asked in exams. RD Sharma Solutions provide clear, step-by-step explanations to problems, helping students grasp the concepts and apply them to solve questions effectively in exams.
3. Real-World Applications: The concepts learned in this chapter have real-world applications in fields such as engineering, architecture, and design. Whether it's calculating the area of a circular garden, understanding the principles behind wheels, or working with circular shapes in architecture, knowledge of circles is crucial.
4. Builds Problem-Solving Skills: RD Sharma Solutions offer various practice problems that help students improve their problem-solving skills. The chapter covers both theoretical and practical questions related to the properties of circles, helping students develop logical reasoning and accuracy in calculations.
5. Prepares for Advanced Geometry: The knowledge gained in this chapter forms the basis for understanding more advanced geometry concepts such as tangents, secants, and cyclic quadrilaterals in higher classes. Mastering the basics of circles allows students to easily move on to more complex topics.
6. Interactive Learning: RD Sharma Solutions are designed to help students learn interactively. With a wide range of questions, students can practice different types of problems and deepen their understanding of the topic. The solutions are structured in a way that students can follow the step-by-step approach to solving problems.
7. Confidence Boost for Students: By practicing all the problems in the chapter and referring to RD Sharma Solutions, students gain confidence in their ability to solve problems related to circles. This not only improves their academic performance but also boosts their overall confidence in tackling mathematical problems.
8. Helps with Comprehensive Revision: The solutions allow students to revise the entire chapter thoroughly, ensuring they don't miss out on any important concepts or formulas. The solutions are aligned with the CBSE syllabus, making it easier for students to prepare for their board exams and assessments.
Conclusion
In summary, RD Sharma Solutions for Class 9 Chapter 16 - Circles is an invaluable resource for mastering circle-related concepts in geometry. By providing a clear understanding of the topic, boosting problem-solving skills, and preparing students for exams and real-world applications, this chapter plays a vital role in a student's academic success in mathematics.
RD Sharma Solutions for Class 9 Maths Chapter 16 Circles FAQs
Why should we refer to RD Sharma Solutions for Class 9 Maths Chapter 16?
RD Sharma Solutions for Class 9 Maths Chapter 16 are an excellent resource because they provide step-by-step solutions to all the exercises in the chapter. These solutions help clarify concepts such as the center, radius, diameter, and other properties of circles. They offer detailed explanations, making it easier for students to understand complex topics and prepare for exams.
What are the topics covered under RD Sharma Solutions for Class 9 Maths Chapter 16?
RD Sharma Solutions for Chapter 16 cover the following topics:
Definition and properties of a circle
Center, radius, diameter, and circumference of a circle
Chords, arcs, and sectors
Area and perimeter of circles
Problems related to tangents, angles, and lengths of arcs
Real-life applications of circle properties
What topics are covered in RD Sharma Class 9 Maths Chapter 16?
Chapter 16 of RD Sharma Class 9 Maths includes the following topics:
Basic definitions and properties of circles
Radius, diameter, and circumference
Understanding of chords, sectors, and segments
Length of an arc and area of sectors
Practical problems involving circles, such as calculating areas, radii, and lengths
How can RD Sharma Solutions help with Chapter 16?
RD Sharma Solutions for Chapter 16 help by breaking down complex circle-related problems into simpler, manageable steps. The solutions provide clear explanations for each topic, helping students build a strong understanding of the subject. Practicing these solutions improves problem-solving skills, and prepares students for their exams by reinforcing core concepts in geometry.
Are RD Sharma Solutions suitable for self-study?
Yes, RD Sharma Solutions are highly suitable for self-study. The detailed, step-by-step explanations and solutions allow students to understand the material on their own. Students can practice problems independently and refer to the solutions when they need clarification, making it an excellent tool for self-paced learning.