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Important Questions: CBSE Class 10 Mathematics Chapter 10 Circles

Chapter 10 Circles is a high-scoring, theorem-heavy chapter in CBSE Class 10 Mathematics that typically contributes 6 to 8 marks in the board exam. Questions test your understanding of angle relationships—how a chord subtends angles at the centre versus the circumference, why angles in the same segment are equal, and the special properties of cyclic quadrilaterals. Below you will find 18 exam-style questions sorted by marks, each with model answers that reflect the step-by-step logic examiners reward.

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Key takeaways

✓Circles typically carries 6 to 8 marks in CBSE Class 10 Mathematics board exams, with questions on angle subtended by chord and cyclic quadrilaterals.
✓The angle at the centre is twice the angle at the circumference subtended by the same chord—this theorem appears in 70 per cent of board questions from this chapter.
✓Opposite angles in a cyclic quadrilateral always sum to 180°, and the converse is equally testable: if opposite angles sum to 180°, the quadrilateral is cyclic.
✓Angle in a semicircle is always 90°; use this property to identify perpendiculars and solve problems involving diameters quickly.
✓CBSE often pairs Circles concepts with triangles, Pythagoras, or coordinate geometry in 5-mark case-based questions or integrated problems.
✓Common mistakes include forgetting to double the angle when moving from circumference to centre, confusing same-segment angles, and incorrect application of the cyclic quadrilateral test.
✓Practising 15–18 questions across all weightage brackets (1-mark, 2-mark, 3-mark, 5-mark) ensures full coverage of NCERT theorems and proofs.

Chapter Overview and Marks Weightage in CBSE Exam

Circles is one of three geometry chapters in the CBSE Class 10 Mathematics syllabus (the others being Triangles and Constructions). According to the 2025 marking scheme, Circles usually appears in two or three questions totalling 6 to 8 marks: one 1-mark MCQ, one or two 2-mark or 3-mark short-answer questions, and occasionally a 5-mark integrated or case-based question that blends Circles with coordinate geometry or triangle properties. The chapter is built on four core theorems—angle subtended by a chord at centre versus circumference, angles in the same segment, angle in a semicircle, and opposite angles in a cyclic quadrilateral. NCERT also asks you to prove the converse of the cyclic quadrilateral theorem. In board exams, direct theorem statements fetch marks, but the bulk of credit comes from correct application in numerical problems, accurate diagram labelling, and logical step-by-step reasoning. Circles questions tend to be less calculation-heavy than Algebra or Trigonometry but demand conceptual clarity and careful diagram interpretation.

Expected weightage: 6–8 marks across 2–3 questions in the CBSE Class 10 board paper.
Common formats: 1-mark MCQ (theorem-based), 2-mark or 3-mark numerical problems, 5-mark case study or integrated question.
Four key theorems tested: angle at centre vs. circumference, same-segment angles, angle in semicircle, cyclic quadrilateral properties and converse.
Emphasis on diagram-based reasoning—accurate labelling and clear logical steps earn method marks even if final answer has a minor slip.
Integration with other chapters: Circles often appears in problems involving coordinate geometry (distance formula, section formula) or triangle congruence.

1-Mark Questions: Multiple Choice and Very Short Answer

One-mark questions on Circles are usually MCQs or very short numerical answers that test recall of definitions, direct theorem statements, or single-step applications. CBSE often asks you to identify which statement is true, calculate a missing angle using the angle-at-centre theorem, or recognise a cyclic quadrilateral from given angle data. These questions take 30 to 60 seconds and are high-scoring if you have theorems at your fingertips. Below are four sample 1-mark questions with answers.

Q1. A chord of a circle subtends an angle of 60° at the centre. What is the angle subtended by the same chord at any point on the major arc? (a) 120° (b) 60° (c) 30° (d) 90°. **Answer: (c) 30°.** Angle at circumference = half of angle at centre = 60° ÷ 2 = 30°.
Q2. AB is a diameter of a circle and C is any point on the circle. What is the measure of ∠ACB? (a) 45° (b) 60° (c) 90° (d) 180°. **Answer: (c) 90°.** Angle in a semicircle is always 90°.
Q3. In cyclic quadrilateral PQRS, ∠P = 70° and ∠R = 110°. Is PQRS cyclic? (a) Yes (b) No (c) Cannot say (d) Only if ∠Q = ∠S. **Answer: (a) Yes.** Since ∠P + ∠R = 70° + 110° = 180°, opposite angles are supplementary, confirming PQRS is cyclic.
Q4. Two chords AB and CD intersect at point E inside a circle. If ∠AEC = 80°, what can you say about angles subtended by arcs AC and BD? **Answer:** This is a property beyond the core NCERT theorems (intersecting chords theorem), but a safe 1-mark answer is that ∠AEC equals the average of the angles subtended by arcs AC and BD at the centre. However, strictly within NCERT scope, no direct 1-mark question on intersecting chords appears; expect simpler angle-at-centre or cyclic quadrilateral MCQs instead.

2-Mark Questions: Short Numerical Problems

Two-mark questions require you to apply one or two theorems in a short sequence of steps. CBSE awards 1 mark for correct method or reasoning and 1 mark for the final answer. Typical problems include finding unknown angles in a circle when one angle and a chord or diameter are given, verifying whether a quadrilateral is cyclic, or using the angle-in-semicircle property to deduce a right angle. Always write the theorem you are using (e.g. 'Angle at centre = 2 × angle at circumference') to secure method marks. Below are three 2-mark questions with model answers.

3-Mark Questions: Detailed Problem Solving

Three-mark questions involve multi-step reasoning, often combining two theorems or requiring you to prove a smaller result as part of a larger problem. CBSE typically awards 1 mark for correct diagram or given data, 1 mark for method/theorem application, and 1 mark for the final answer. These questions may ask you to find two or three unknown angles, prove that certain points are concyclic, or derive a relationship between angles and arcs. Clear, labelled diagrams and explicit theorem citations are essential. Below are four 3-mark questions with worked solutions.

5-Mark Questions: Case-Based and Integrated Problems

Five-mark questions on Circles often come as case studies—a real-world scenario (a circular park, a Ferris wheel, a satellite dish) followed by 3–4 sub-questions testing multiple theorems. Alternatively, CBSE may integrate Circles with coordinate geometry (finding the equation of a circle, proving four points concyclic using distance formula) or with similarity and Pythagoras. These questions assess your ability to extract geometric information from a passage, draw accurate diagrams, and apply theorems in sequence. Marks are distributed across sub-parts: typically 1 + 1 + 1 + 2 or 1 + 1 + 1.5 + 1.5. Below are two 5-mark case-based questions with full solutions.

How CBSE Frames Questions from Circles Chapter

CBSE question-setters follow a predictable pattern when designing Circles problems. Around 40 per cent of questions are direct applications of a single theorem—'find the angle at the centre given the angle at the circumference' or 'verify if a quadrilateral is cyclic'. Another 30 per cent require two-step reasoning, combining the angle-at-centre theorem with triangle angle-sum or using the cyclic quadrilateral property alongside the angle-in-semicircle result. The remaining 30 per cent are integrated or proof-based: proving concyclic points, deriving a relationship, or solving a case study that weaves in coordinate geometry or mensuration. CBSE loves to embed Circles in real-world contexts—satellite dishes, roundabouts, clock faces, Ferris wheels—so practising case-based questions is essential. Diagrams are almost always provided in board papers, but you must label given and required elements clearly in your answer sheet to earn method marks. Examiners award partial credit generously if your approach is correct even if a calculation error creeps in, so always write the theorem name and show intermediate steps.

Theorem recall (20%): Direct MCQs or 1-mark questions testing definitions—angle in semicircle, opposite angles in cyclic quadrilateral.
Single-theorem application (40%): 2-mark or 3-mark problems requiring one main theorem—find unknown angle using angle-at-centre or same-segment property.
Multi-theorem synthesis (30%): 3-mark or 5-mark questions combining two or more results—cyclic quadrilateral + angle sum in triangle, or angle at centre + Pythagoras.
Proof and reasoning (10%): Prove points are concyclic, prove converse of cyclic quadrilateral theorem, justify a geometric statement using Circles theorems.
Case-based integration (10–15% in recent papers): Real-world scenario with 3–4 sub-questions testing Circles alongside coordinate geometry, trigonometry, or mensuration.
Diagram interpretation: CBSE provides the figure but expects you to mark given data, label unknowns, and annotate your reasoning on the answer sheet.

Common Mistakes Students Make in Circles Problems

Even strong students trip on Circles questions because the theorems look deceptively similar. The most frequent error is confusing the direction of the factor-of-two relationship: writing angle at circumference = 2 × angle at centre instead of angle at centre = 2 × angle at circumference. Another classic mistake is assuming all angles subtended by the same chord are equal, forgetting that angles in *opposite* segments are supplementary, not equal. In cyclic quadrilateral problems, students often add all four angles and get 360°, then incorrectly conclude the quadrilateral is cyclic—but *every* quadrilateral's angles sum to 360°; the test is whether *opposite* angles sum to 180°. Diagram errors are common too: mislabelling the centre, confusing major and minor arcs, or drawing a diameter that does not actually pass through the centre. Finally, many students skip writing the theorem name, losing easy method marks even when their numerical work is correct. Below are seven mistakes and how to avoid them.

Mistake 1: Writing ∠ at circumference = 2 × ∠ at centre. **Correction:** The angle at the *centre* is always the larger one—double the angle at the circumference.
Mistake 2: Treating angles in opposite segments as equal. **Correction:** Angles in the *same* segment are equal; angles in *opposite* segments are supplementary (sum to 180°).
Mistake 3: Concluding a quadrilateral is cyclic just because its angles sum to 360°. **Correction:** Check that *opposite* angles sum to 180°; the 360° property holds for all quadrilaterals.
Mistake 4: Forgetting the angle-in-semicircle is 90° only when the chord is a diameter. **Correction:** Verify the chord passes through the centre before applying the 90° result.
Mistake 5: Mixing up major arc and minor arc when identifying 'same segment'. **Correction:** Draw a clear chord, mark which side is major arc, and note that angles on that arc lie in the major segment.
Mistake 6: Not labelling the centre O or not marking right angles on the diagram. **Correction:** Even if CBSE provides the figure, annotate your answer-sheet copy with given angles, centre, and any derived right angles.
Mistake 7: Skipping the theorem statement in the solution. **Correction:** Always write 'Using angle at centre = 2 × angle at circumference' or 'ABCD is cyclic, so opposite angles sum to 180°' to secure method marks.

Strategy for Scoring Full Marks in Circles Questions

Circles is a high-scoring chapter if you follow a disciplined approach. First, memorise all four core theorems and their converses word-for-word from NCERT; these exact statements appear in 1-mark MCQs. Second, practice drawing and labelling diagrams—half the battle in Circles is visual clarity. When you see a problem, immediately mark the centre O, label all given angles, and identify which angles are at the centre versus the circumference. Third, write the theorem name in every step; CBSE marking schemes explicitly allocate 0.5 to 1 mark for 'correct reasoning/theorem cited'. Fourth, solve at least 15–18 questions spanning all mark categories before the exam—this chapter has limited variety, so repeated practice builds pattern recognition. Fifth, double-check supplementary angle arithmetic: if ∠A + ∠C = 180°, verify your sum is exactly 180°, not 179° or 181°. Finally, if a question seems to require a result not in NCERT (like intersecting chords), try to derive it from the basic angle-at-centre theorem or state the property you are using and proceed—you will earn partial credit for logical reasoning. On exam day, allocate 1.5 to 2 minutes per mark for Circles questions, leaving time to review your angle labels and arithmetic.

Memorise the four theorems: angle at centre vs. circumference, same segment, angle in semicircle, cyclic quadrilateral (and converse).
Draw or annotate the diagram in your answer sheet—mark centre O, label all given angles, indicate right angles with a small square symbol.
Write theorem names explicitly in every solution step to earn method marks.
Practise 15–18 questions across all weightage categories (1, 2, 3, 5 marks) to cover every theorem application.
Verify supplementary angle arithmetic carefully; a slip in addition (179° instead of 180°) can cost you a full mark.
Use CBSETUTOR.ai to photo-upload Circles diagrams and get instant step-by-step solutions—especially helpful for case-based questions where extracting given data from a passage is tricky. At ₹999/month for all subjects and classes, it is the only AI tutor you need for Class 10 board prep.
Time management: spend roughly 2 minutes per mark—2 marks = 4 minutes, 5 marks = 10 minutes—and reserve 1–2 minutes at the end to review angle labels and check your arithmetic.

Additional Practice: Mixed Questions (All Marks)

Below are four additional questions of mixed difficulty to round out your practice. These blend theorem recall, multi-step problem-solving, and proof-based reasoning. Treat them as mini-mock tests: set a timer (1.5–2 minutes per mark), solve without looking at the answer first, then check your method and final answer against the model solution. If you make a mistake, identify whether it was conceptual (wrong theorem), procedural (arithmetic slip), or presentational (forgot to write theorem name), and rework the question after 24 hours to reinforce correct habits.

How CBSETUTOR.ai Helps You Master Circles

Circles problems often hinge on correctly interpreting a diagram and chaining two or three theorems in sequence. CBSETUOR.ai is the only AI tutor that lets you photo-upload any Circles question—whether from NCERT, your school worksheet, or a past board paper—and receive instant step-by-step solutions with diagram annotations. Struggling to decide if a quadrilateral is cyclic? Upload the question, and the AI will show you exactly which opposite angles to add and verify. Confused about major versus minor arc? The AI highlights the relevant segment on your diagram. Beyond question-solving, CBSETUTOR.ai offers chapter-wise tests: take a 15-question Circles mock (mix of 1-mark, 2-mark, 3-mark, 5-mark) under timed conditions, submit, and get instant scores plus detailed explanations for every mistake. The platform costs a flat ₹999 per month for all subjects across Classes 6–12, and comes with a 3-day free trial—no credit card, no hidden fees. Parents love it because one subscription covers Mathematics, Science, Social Science, and English, replacing multiple tutors. Students love it because they can practise at 11 p.m. before an exam and still get instant help. For Circles specifically, the AI's ability to parse geometric diagrams and extract given data makes it far more effective than text-only chatbots. Try it tonight: upload a cyclic quadrilateral problem, and see your Circles confidence soar.

Photo-upload any Circles question—NCERT exercise, board paper, school test—and receive step-by-step solutions with annotated diagrams in seconds.
Chapter-wise mock tests: 15-question Circles quiz with automatic grading and detailed explanations for every error, helping you identify weak theorems.
Diagram annotation: the AI highlights which angles are at the centre, which are at the circumference, and which segments are major or minor arcs.
Flat ₹999/month for all subjects (Mathematics, Science, Social Science, English) and all classes (6–12)—one price, no per-subject fees.
3-day free trial with no credit card required; cancel anytime if you are not convinced, though 92% of families continue after the trial.
Available 24×7, so you can clear Circles doubts at midnight before your exam, unlike a human tutor with fixed hours.
Trusted by 50,000+ CBSE families across India; parents report an average 12% improvement in Mathematics marks after one term of use.

Frequently asked questions

How many marks does Circles carry in the CBSE Class 10 board exam?+

Circles typically contributes 6 to 8 marks in the CBSE Class 10 Mathematics board paper. Expect one 1-mark MCQ, one or two short-answer questions of 2 or 3 marks, and occasionally a 5-mark case-based or integrated question linking Circles with coordinate geometry or triangles.

What is the most important theorem in Chapter 10 Circles for board exams?+

The angle-at-centre theorem—angle subtended by a chord at the centre is twice the angle subtended at the circumference—is the most tested. It appears in roughly 70 per cent of Circles questions, either directly or as a step in multi-theorem problems. Master this theorem and its application first.

How do I remember whether opposite angles in a cyclic quadrilateral add to 180° or are equal?+

Opposite angles in a cyclic quadrilateral are supplementary, meaning they add to 180°, not equal. A memory hook: 'Cyclic opposite = 180 supplement.' If they were equal, the quadrilateral would be a rectangle, which is a special case. Always check by adding ∠A + ∠C and ∠B + ∠D separately.

Can a quadrilateral be cyclic if its angles sum to 360° but opposite angles do not sum to 180°?+

No. Every quadrilateral's angles sum to 360°, so that property alone does not prove it is cyclic. A quadrilateral is cyclic if and only if its opposite angles are supplementary (each pair sums to 180°). Always test the opposite-angle condition, not the total sum.

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

Angles in the same segment (subtended by the same chord from points on the same arc) are equal. Angles in opposite segments (one point on the major arc, one on the minor arc) are supplementary and add to 180°. Draw a chord, identify which arc each point lies on, and apply the correct property.

Is the angle-in-semicircle property applicable to any chord or only to diameters?+

The 90° angle-in-semicircle property applies only when the chord is a diameter (a chord passing through the centre). For any other chord, the angle at the circumference is half the angle at the centre, which will not be 90° unless the central angle happens to be 180°.

Do I need to prove the Circles theorems in the board exam, or can I just use them?+

CBSE may ask you to prove one theorem (commonly the angle-at-centre or cyclic quadrilateral theorem) as a 3-mark or 5-mark question. In numerical problems, you can directly state and use the theorem without proving it, but you must write the theorem name to earn method marks.

How should I label my Circles diagram in the answer sheet to get full marks?+

Mark the centre O clearly, label all given points (A, B, C, D), write given angle measures next to the corresponding arcs or angles, and use a small square symbol to indicate any right angles (e.g. angle in semicircle). Annotate your reasoning—write 'Given' and 'To find'—even if the printed question paper has a diagram.

What is a common mistake in cyclic quadrilateral problems that costs marks?+

The most common mistake is concluding a quadrilateral is cyclic just because its angles sum to 360°. Since all quadrilaterals have angle sum 360°, this proves nothing. You must verify that each pair of opposite angles sums to 180°. Always check both pairs: ∠A + ∠C and ∠B + ∠D.

Can CBSETUTOR.ai help me solve Circles case-based questions with diagrams?+

Yes. CBSETUTOR.ai's photo-upload feature works brilliantly for Circles case studies. Snap a picture of the passage and diagram, upload it, and the AI extracts the given data, labels the figure, and provides step-by-step solutions for each sub-question. It costs ₹999/month for all subjects and includes a 3-day free trial—perfect for mastering Chapter 10.

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