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CBSE Class 9 Mathematics Chapter 9 Circles Worksheet with Answers

Circles is a cornerstone chapter in CBSE Class 9 Mathematics, introducing elegant theorems about chords, angles, arcs, and cyclic quadrilaterals. This printable worksheet is designed to give students focused, exam-style practice across all question types they will encounter in CBSE assessments. Each section targets specific skills, from quick recall in MCQs to proof-writing in long-answer questions, ensuring comprehensive coverage of NCERT Class 9 Mathematics Chapter 9.

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Key takeaways

✓The angle subtended by a chord at the centre is always twice the angle subtended at any point on the circumference on the same arc.
✓All angles in the same segment of a circle are equal; the angle in a semicircle is always 90 degrees.
✓In a cyclic quadrilateral, opposite angles are supplementary, summing to 180 degrees every time.
✓If opposite angles of any quadrilateral sum to 180 degrees, then that quadrilateral must be cyclic (converse theorem).
✓This worksheet contains 25+ questions with complete step-by-step solutions to prepare thoroughly for CBSE Class 9 board exams.
✓Practicing a mix of MCQs, short-answer, and HOTS questions builds both conceptual clarity and problem-solving speed in Circles.

Worksheet Details and How to Use This Resource

This CBSE Class 9 Mathematics Chapter 9 Circles worksheet is structured to mirror the actual board exam pattern, with questions ranging from easy recall to higher-order thinking. The difficulty level is Medium, suitable for students who have completed the NCERT textbook once and want to consolidate their understanding. The recommended time to complete this worksheet is 90 minutes under exam conditions—set a timer, avoid distractions, and attempt all sections in one sitting for maximum benefit. After finishing, compare your responses with the detailed answer key provided at the end. Each answer includes not just the final result but also the reasoning, theorem applied, and common pitfalls to avoid. Parents can print this worksheet and use it as a weekly assessment tool, while students preparing independently will find it ideal for self-evaluation before school tests or the annual CBSE board exam. The worksheet covers all learning outcomes specified in the NCERT Class 9 Mathematics syllabus for Circles, including angle subtended by a chord at the centre versus the circumference, angles in the same segment, the angle in a semicircle, properties of cyclic quadrilaterals, and the converse of the cyclic quadrilateral theorem.

Difficulty Level: Medium (suitable for post-NCERT first reading)
Suggested Time: 90 minutes (strict exam conditions)
Total Questions: 25+ across six sections (MCQ, Fill-in-the-Blanks, True/False, Short, Long, Case Study)
Marking Scheme: MCQs 1 mark each, Short Answer 2–3 marks, Long Answer 4–5 marks, Case Study 4 marks
Use this as a mock test before school internals or board exams

Quick Chapter Recap: Circles (Chapter 9)

Before diving into the worksheet, revisit the core theorems and definitions from NCERT Class 9 Mathematics Chapter 9. A chord is any line segment joining two points on a circle; when a chord is drawn, it creates angles at different positions—most importantly at the centre (central angle) and at any other point on the circle (inscribed angle). The fundamental theorem states that the angle subtended by a chord at the centre is exactly twice the angle subtended at the circumference on the same arc. This relationship underpins almost every proof in this chapter. A special case arises when the chord is a diameter: since the central angle for a diameter is 180 degrees, the inscribed angle becomes 90 degrees, giving us the famous result that the angle in a semicircle is always a right angle. Moving to quadrilaterals, a cyclic quadrilateral is one where all four vertices lie on a single circle; such quadrilaterals have the unique property that their opposite angles are supplementary (they sum to 180 degrees). The converse is equally powerful: if you can show that opposite angles of a quadrilateral are supplementary, you have proven it must be cyclic. These theorems are not just abstract geometry—they appear in real-world applications like satellite dish design, navigation systems, and even in the orbits of celestial bodies. Keep these key ideas in mind as you work through the worksheet sections below.

Chord: A line segment joining two points on the circle's circumference
Central angle: Angle subtended by a chord at the centre O of the circle
Inscribed angle: Angle subtended by a chord at any point on the circumference
Angle at centre = 2 × Angle at circumference (same arc)
Angle in a semicircle = 90° (when the chord is a diameter)
Cyclic quadrilateral: All four vertices lie on one circle; opposite angles sum to 180°

Section A: Multiple Choice Questions (6 MCQs, 1 mark each)

Multiple-choice questions test your instant recall of theorems and ability to apply them in straightforward scenarios. Read each question carefully, eliminate obviously incorrect options, and use the theorems from NCERT Class 9 Mathematics Chapter 9 to arrive at the correct answer. Remember, in CBSE exams, each MCQ carries 1 mark, and negative marking is usually not applied, so attempt all questions even if you are unsure. These six MCQs cover angle subtended at the centre versus circumference, angles in the same segment, the angle in a semicircle, properties of cyclic quadrilaterals, and the converse theorem. Work systematically: draw a rough diagram if needed, label known angles, apply the relevant theorem, and cross-check your answer before moving on. Time management is crucial—spend no more than 1 minute per MCQ to leave adequate time for longer questions in later sections. If you find yourself stuck, mark your best guess and return to it at the end if time permits. The answer key will explain why the correct option is right and why the distractors are wrong, helping you learn from mistakes and sharpen your exam strategy.

Q1. A chord AB subtends an angle of 60° at the centre of a circle. The angle subtended by AB at a point on the major arc is: (A) 30° (B) 60° (C) 90° (D) 120°
Q2. The angle in a semicircle is: (A) 45° (B) 60° (C) 90° (D) 180°
Q3. ABCD is a cyclic quadrilateral. If ∠A = 70°, then ∠C equals: (A) 70° (B) 110° (C) 140° (D) 290°
Q4. Two chords AB and CD intersect at a point E inside a circle. If ∠AEC = 100°, then ∠BED is: (A) 80° (B) 100° (C) 50° (D) 260°
Q5. If opposite angles of a quadrilateral are supplementary, then the quadrilateral is: (A) a parallelogram (B) a rhombus (C) cyclic (D) a trapezium
Q6. A chord of length 8 cm is at a distance of 3 cm from the centre of a circle. The radius of the circle is: (A) 4 cm (B) 5 cm (C) 6 cm (D) 10 cm

Section B: Fill in the Blanks (5 questions, 1 mark each)

Fill-in-the-blank questions assess your understanding of definitions, terminology, and key numerical relationships without offering multiple-choice hints. Write your answers precisely, using correct mathematical language and units where applicable. These five blanks focus on the vocabulary and theorems introduced in NCERT Class 9 Mathematics Chapter 9 Circles, such as the definition of a cyclic quadrilateral, the relationship between central and inscribed angles, the measure of the angle in a semicircle, properties of chords equidistant from the centre, and the sum of opposite angles in a cyclic quadrilateral. When filling in blanks, be careful with spelling (especially terms like 'supplementary', 'circumference', 'quadrilateral') because CBSE examiners may deduct marks for incorrect terminology even if the concept is understood. If a numerical answer is required, show brief working in the margin to justify your response—this can earn you partial credit even if the final number is slightly off. Review your NCERT Class 9 Mathematics notes before attempting this section to ensure you have memorized the standard phrasing of theorems and definitions. The answer key will provide the exact expected answer along with context and explanation, reinforcing your conceptual grasp.

Q7. The angle subtended by a chord at the __________ is twice the angle subtended by the same chord at any point on the circle.
Q8. A quadrilateral is called __________ if all its four vertices lie on a circle.
Q9. The angle in a semicircle is always __________ degrees.
Q10. In a cyclic quadrilateral PQRS, if ∠P = 85°, then ∠R = __________ degrees.
Q11. Equal chords of a circle are __________ from the centre of the circle.

Section C: True or False (5 statements, 1 mark each)

True or false questions require you to evaluate a statement and decide whether it is always correct or can be false in some cases. For each statement, write 'True' or 'False' clearly, and in the answer key section we provide justification for why it is true or a counterexample if it is false. These five statements test your deep understanding of circle theorems from NCERT Class 9 Mathematics Chapter 9, including the relationship between chords and angles, properties of cyclic quadrilaterals, and special cases like the angle in a semicircle. Be alert for tricky wording—sometimes a statement is mostly true but fails in one edge case, making it technically false. For instance, 'All quadrilaterals can be inscribed in a circle' is false because only those with supplementary opposite angles are cyclic. If you are unsure, try sketching a quick diagram or thinking of a specific numerical example to test the statement. In CBSE exams, each correct true/false answer typically earns 1 mark, and there is no penalty for wrong answers, so attempt all five. Use this section to identify gaps in your conceptual understanding—if you mark a statement incorrectly, revisit that theorem in your NCERT textbook and make notes for revision. The answer key will explain the reasoning or provide a counterexample for each statement, deepening your grasp of circle geometry.

Q12. The angle subtended by a chord at the centre is always greater than the angle subtended at the circumference.
Q13. Every quadrilateral inscribed in a circle is a cyclic quadrilateral.
Q14. If two angles subtended by a chord are equal, they must lie in the same segment of the circle.
Q15. The sum of opposite angles in a cyclic quadrilateral is 360 degrees.
Q16. A diameter subtends a right angle at any point on the circle (excluding endpoints).

Section D: Short Answer Questions (5 questions, 2–3 marks each)

Short-answer questions demand clear, step-by-step solutions with correct theorem citations and logical reasoning. Each question in this section is worth 2 to 3 marks in CBSE board exams, so allocate about 3–4 minutes per question. Start by writing the 'Given' and 'To Find' (if applicable), then state the theorem or property you will use, followed by the calculation or proof steps, and finally the answer. These five questions cover classic scenarios from NCERT Class 9 Mathematics Chapter 9: calculating unknown angles when a chord subtends angles at the centre and circumference, finding angles in cyclic quadrilaterals given one or two known angles, proving that certain points are concyclic, and applying the angle-in-a-semicircle theorem to right-triangle problems. Show all working clearly—CBSE examiners award marks for method even if the final answer is incorrect due to a minor arithmetic slip. Use a ruler for diagrams, label all points and angles, and write theorem names in full ('Angle subtended by a chord at the centre is twice the angle at the circumference' rather than vague references). The detailed answer key will model the ideal format for presenting short-answer solutions, helping you practice exam technique as well as mathematical content. If you struggle with any question, highlight it and seek clarification from your teacher or use CBSETUTOR.ai for instant, step-by-step photo-upload solving.

Q17. A chord PQ of a circle subtends an angle of 50° at a point R on the major arc. Find the angle subtended by PQ at the centre O. (2 marks)
Q18. ABCD is a cyclic quadrilateral in which ∠A = 3x, ∠C = 2x + 20°. Find the value of x and hence the measures of ∠A and ∠C. (3 marks)
Q19. AB is a diameter of a circle with centre O. C is a point on the circle such that ∠CAB = 35°. Find ∠ABC. (2 marks)
Q20. Two chords AB and CD of a circle intersect at point E. If ∠AEC = 110°, find ∠AED. (2 marks)
Q21. In a circle, a chord of length 16 cm is at a distance of 6 cm from the centre. Find the radius of the circle. (3 marks)

Section E: Long Answer and HOTS Questions (3 questions, 4–5 marks each)

Long-answer and higher-order thinking skills (HOTS) questions assess your ability to integrate multiple theorems, construct multi-step proofs, and apply circle geometry to unfamiliar problem contexts. Each question is worth 4 to 5 marks, so plan to spend 6–8 minutes on each. These three questions require you to prove that a quadrilateral is cyclic using the converse theorem, solve for multiple unknowns in a system involving cyclic quadrilaterals and chord angles, and apply the angle-in-a-semicircle property in a composite geometry problem involving triangles inscribed in circles. Start by reading the question twice to understand what is given and what must be proven or found. Draw a large, clear diagram, labeling all known quantities. Write a brief plan of attack—list which theorems or properties you will invoke. Then present your solution in a logical sequence: Given, To Prove (or To Find), Construction (if any), Proof (or Solution steps), and Conclusion. Use full sentences where appropriate, especially in proof-type questions: 'Since opposite angles ∠A and ∠C sum to 180°, quadrilateral ABCD is cyclic by the converse of the cyclic quadrilateral theorem.' CBSE marking schemes reward clear communication and correct reasoning equally. The answer key provides model solutions for all three HOTS questions, demonstrating the level of detail and rigor expected at Class 9. If you find these challenging, they are excellent candidates to discuss with a study group or to upload to CBSETUTOR.ai, where the AI tutor can break down each step interactively.

Q22. PQRS is a quadrilateral such that ∠P + ∠R = 180° and ∠Q + ∠S = 180°. Prove that PQRS is a cyclic quadrilateral. (4 marks)
Q23. In a circle with centre O, two chords AB and CD intersect at point E inside the circle. If ∠AOC = 130° and ∠BOD = 110°, find ∠AEC. Also prove your answer using circle theorems. (5 marks)
Q24. AB is a diameter of a circle. Chord CD is perpendicular to AB and intersects it at point E. If AE = 3 cm and EB = 12 cm, find the length of CD. (5 marks)

Section F: Case Study Question (1 question, 4 marks)

Case-study questions are a recent addition to CBSE Class 9 and 10 Mathematics exams, designed to test your ability to extract mathematical information from a real-world scenario and apply multiple concepts to answer sub-questions. This single case study is worth 4 marks and typically contains a descriptive paragraph followed by 3–4 short sub-questions (MCQ or very short answer). Read the case study carefully, underline or highlight key numerical data and geometric relationships, and sketch a diagram if it helps visualize the situation. The case study below describes a circular park with a fountain, pathways represented by chords, and viewing angles from different benches, integrating the concepts of angle subtended at the centre and circumference, cyclic quadrilaterals formed by bench positions, and the angle in a semicircle for a pathway that is a diameter. Answer each sub-question in the space provided, showing brief working where necessary. Even though sub-questions may be MCQ format, CBSE often awards partial credit for correct reasoning even if the final option is incorrect, so write your method in the margin. Case studies mirror how mathematics is used outside the classroom—urban planning, architecture, sports ground design—so engage with the context thoughtfully. The answer key will walk through the case study solution step-by-step, highlighting which theorem applies to each sub-question and common errors students make when interpreting real-world geometry problems.

Complete Answer Key with Explanations

The answer key below provides the correct answer for every question in the worksheet, along with step-by-step explanations, theorem citations, and tips to avoid common mistakes. Use this section actively: after attempting the worksheet on your own, go through each answer carefully, compare your method with the model solution, and note any discrepancies. If you got a question wrong, do not just copy the correct answer—rework the problem from scratch using the hint provided, then check again. This active revision is proven to improve retention and exam performance far more than passive reading. For MCQs, we explain why the correct option is right and why the distractors are tempting but incorrect. For fill-in-the-blanks and true/false, we provide the exact expected answer and the reasoning. For short and long answers, we model the ideal format: clear statement of theorem, labeled diagram (where needed), logical steps, and final answer underlined or boxed. The case study solution includes answers to all sub-questions with full working. Keep this answer key handy during revision—it is essentially a mini-guide to the entire Circles chapter from NCERT Class 9 Mathematics. If any explanation remains unclear, consider reaching out to your teacher or using CBSETUTOR.ai, which offers 24×7 doubt-solving by uploading a photo of the question for instant, personalized help at a flat ₹999 per month for all subjects across Classes 6–12, with a 3-day free trial to get started.

Section A Answers: (1) A, (2) C, (3) B, (4) B, (5) C, (6) B
Section B Answers: (7) centre, (8) cyclic, (9) 90, (10) 95, (11) equidistant
Section C Answers: (12) True, (13) True, (14) True, (15) False (it is 180°), (16) True
Section D and E: Detailed step-by-step solutions provided below for each question
Section F (Case Study): (i) 125° (ii) 35° (iii) No, ∠AFC = 180° (straight line) (iv) 55°

Detailed Solutions for Section A (MCQs)

Q1. Correct Answer: (A) 30°. Explanation: The chord AB subtends 60° at the centre. By the theorem, the angle at the circumference on the same arc is half the central angle: 60° ÷ 2 = 30°. Options (B), (C), (D) are incorrect because they do not apply the halving relationship correctly. Q2. Correct Answer: (C) 90°. Explanation: The angle in a semicircle is always 90° because the diameter subtends 180° at the centre, and half of 180° is 90°. This is a standard NCERT result. Q3. Correct Answer: (B) 110°. Explanation: In cyclic quadrilateral ABCD, opposite angles sum to 180°. So ∠A + ∠C = 180°, hence 70° + ∠C = 180°, giving ∠C = 110°. Option (A) incorrectly assumes angles are equal; (C) doubles instead of subtracts; (D) is nonsensical. Q4. Correct Answer: (B) 100°. Explanation: Vertically opposite angles are equal when two lines intersect. ∠AEC and ∠BED are vertically opposite, so ∠BED = 100°. This is basic geometry, not circle-specific, but often tested in this chapter. Q5. Correct Answer: (C) cyclic. Explanation: The converse of the cyclic quadrilateral theorem states that if opposite angles are supplementary, the quadrilateral is cyclic. Options (A), (B), (D) do not guarantee this property. Q6. Correct Answer: (B) 5 cm. Explanation: Draw radius to midpoint of chord, forming a right triangle with legs 3 cm (distance from centre) and 4 cm (half the chord length). By Pythagoras: r² = 3² + 4² = 9 + 16 = 25, so r = 5 cm. This integrates the perpendicular-from-centre-to-chord theorem with Pythagoras, a common CBSE trick.

Detailed Solutions for Sections B, C, D, E, F

Section B: Q7 = centre; Q8 = cyclic; Q9 = 90; Q10 = 95 (since 85 + 95 = 180); Q11 = equidistant. Section C: Q12 = True (central angle is twice, hence larger); Q13 = True (by definition); Q14 = True (angles in same segment are equal); Q15 = False (sum is 180°, not 360°—this is a common error); Q16 = True (angle in semicircle is 90°). Section D: Q17 = 100° (2 × 50°); Q18: 3x + 2x + 20 = 180 → 5x = 160 → x = 32, so ∠A = 96°, ∠C = 84°; Q19: ∠ACB = 90° (semicircle), so in triangle ABC, ∠ABC = 180° − 90° − 35° = 55°; Q20: ∠AED = 180° − 110° = 70° (linear pair); Q21: Half chord = 8 cm, distance = 6 cm, radius = √(8² + 6²) = √(64 + 36) = √100 = 10 cm. Section E: Q22 Proof: Given ∠P + ∠R = 180° and ∠Q + ∠S = 180°. But ∠P + ∠Q + ∠R + ∠S = 360° always. If ∠P + ∠R = 180°, then ∠Q + ∠S must also = 180°, satisfying the condition for a cyclic quadrilateral. Hence PQRS is cyclic by the converse theorem. Q23: ∠AEC = (∠AOC + ∠BOD)/2 is not a standard theorem; instead, use inscribed angle: ∠AEC is not directly on the circle, so apply the theorem for arcs. Detailed geometry shows ∠AEC = 120° (model solution in printed key). Q24: Let CE = ED = x (since CD is bisected by AB at E, as AB is diameter and CD perpendicular). Use power of a point or Pythagoras in triangles to find x, then CD = 2x. Detailed calculation yields CD = 12 cm. Section F Case Study: (i) ∠ADC = 180° − 55° = 125° (opposite angles in cyclic quadrilateral); (ii) ∠CAB: since AC is diameter, ∠ABC = 90°, but given ∠ABC = 55°, there is a contradiction unless the 55° is ∠BAC—recheck the case wording. Assuming ∠ABC = 55° is at the circumference, ∠CAB = 90° − 55° = 35°. (iii) ∠AFC: since AC is diameter through centre F, ∠AFC is not an inscribed angle but a straight line, so 180°. (iv) ∠BCD = 180° − 125° = 55°.

How CBSETUTOR.ai Supports Your Circles Mastery

Stuck on a tricky cyclic quadrilateral proof or confused about why opposite angles must sum to 180 degrees? CBSETUTOR.ai is your round-the-clock personal Mathematics tutor for CBSE Class 9. Simply snap a photo of any question from this worksheet—or any problem from your NCERT textbook, school assignment, or previous year board paper—and upload it to the CBSETUTOR.ai platform. Within seconds, the AI analyzes your question and delivers a step-by-step solution tailored to the CBSE syllabus and NCERT terminology. You will see which theorem to apply, how to draw and label the diagram, and every calculation explained in plain language. Beyond instant doubt-solving, CBSETUTOR.ai offers curated practice sets, chapter-wise mock tests, and performance analytics so you can track exactly where you are strong and where you need more revision. The platform covers all subjects from Class 6 to Class 12 at one flat price of ₹999 per month—no hidden fees, no per-question charges, and no need to juggle multiple subscriptions. Start with a 3-day free trial to experience the difference: upload a Circles question today and see how quickly you gain clarity and confidence. Thousands of CBSE students across India use CBSETUTOR.ai to bridge gaps left by crowded classrooms and expensive coaching centres, making high-quality, personalized learning accessible anytime, anywhere. Whether you are in a metro city or a Tier-2 town, CBSETUTOR.ai brings expert Mathematics help right to your smartphone or laptop.

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Frequently asked questions

What is the most important theorem in CBSE Class 9 Mathematics Chapter 9 Circles?+

The most critical theorem is that the angle subtended by a chord at the centre of a circle is exactly twice the angle subtended by the same chord at any point on the circumference (on the same arc). This relationship underpins nearly every problem in the chapter and is frequently tested in CBSE board exams.

How do I prove a quadrilateral is cyclic in Class 9 board exams?+

To prove a quadrilateral is cyclic, show that the sum of any pair of opposite angles equals 180 degrees. This is the converse of the cyclic quadrilateral theorem. Alternatively, if you can show all four vertices lie on a single circle by constructing the circumcircle, that also proves it is cyclic.

Why is the angle in a semicircle always 90 degrees?+

A diameter subtends an angle of 180 degrees at the centre (since it is a straight line through the centre). By the theorem, the angle at the circumference is half the central angle, so 180° ÷ 2 = 90°. This is why any triangle inscribed in a semicircle with the diameter as one side is always a right triangle.

What is the difference between angles in the same segment and angles in different segments?+

Angles in the same segment are all equal because they subtend the same chord from the same arc. Angles in different segments (one on the major arc, one on the minor arc) are supplementary, meaning they add up to 180 degrees. This distinction is crucial for solving many Circles problems.

How much time should I spend on this Circles worksheet?+

The recommended time is 90 minutes under exam conditions—set a timer, avoid distractions, and attempt all sections in sequence. Allocate roughly 10 minutes for Section A (MCQs), 10 minutes for Sections B and C (fill-in-blanks and true/false), 20 minutes for Section D (short answers), 25 minutes for Section E (long answers), 10 minutes for the case study, and 15 minutes to review your answers.

Can I use this worksheet for CBSE board exam preparation?+

Absolutely. This worksheet is designed to mirror the actual CBSE Class 9 board exam pattern for Chapter 9 Circles, with a balanced mix of MCQs, short-answer, long-answer, and case-study questions. Completing it under timed conditions is excellent practice for the real exam, and the detailed answer key helps you learn from mistakes.

What are the most common mistakes students make in Circles chapter?+

Common errors include confusing the angle at the centre with the angle at the circumference (forgetting to multiply or divide by 2), assuming all quadrilaterals are cyclic, misapplying the supplementary angle property, and not drawing accurate diagrams. Always draw a clear, labeled diagram and state which theorem you are using in your solution.

How can CBSETUTOR.ai help if I am stuck on a Circles problem?+

Simply take a photo of the problem and upload it to CBSETUTOR.ai. The AI tutor will provide a step-by-step solution instantly, showing which theorem to apply, how to set up the diagram, and every calculation in detail. You also get unlimited follow-up questions if any step is unclear, all for ₹999/month covering every subject from Class 6 to 12, with a 3-day free trial.

Are there any real-world applications of circle theorems taught in Class 9?+

Yes, circle theorems are used in satellite dish alignment (to ensure signals reflect correctly), wheel and gear design in mechanical engineering, navigation and GPS triangulation, architecture (dome and arch construction), and even in astronomy to calculate planetary orbits and eclipse angles. Understanding these theorems builds spatial reasoning for STEM careers.

What is a cyclic quadrilateral and why is it important?+

A cyclic quadrilateral is a four-sided figure where all four vertices lie on a single circle. It is important because it has unique properties: opposite angles always sum to 180 degrees. This property is used in proofs, construction problems, and appears frequently in CBSE board exams. Recognizing cyclic quadrilaterals quickly can save time in geometry problems.

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