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CBSE Class 10 Mathematics Chapter 10 Circles Worksheet with Answers

Chapter 10 Circles is a foundation of CBSE Class 10 Mathematics geometry, contributing 8-10 marks in the board exam. This printable worksheet is tailored for 90-minute timed practice, covering all core concepts: angle subtended by a chord at the centre versus the circumference, angles in the same segment, angle in a semicircle, and properties of cyclic quadrilaterals. Each section—from MCQs to case-based questions—mirrors the latest CBSE exam blueprint, ensuring students build speed, accuracy, and conceptual depth.

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Key takeaways

✓The angle subtended by a chord at the centre is exactly twice the angle subtended at any point on the circumference on the same arc.
✓Any angle inscribed in a semicircle (angle subtended by a diameter at the circumference) is always a right angle of 90 degrees.
✓Angles in the same segment of a circle are equal, meaning any two angles subtended by the same chord from points on the same arc are congruent.
✓In a cyclic quadrilateral, opposite angles are supplementary and always add up to 180 degrees, a property unique to quadrilaterals inscribed in circles.
✓The converse is equally powerful: if the sum of opposite angles in any quadrilateral equals 180 degrees, that quadrilateral must be cyclic.
✓Practising structured worksheets with MCQs, case studies, and HOTS questions builds the deep conceptual clarity needed for scoring 35+ in CBSE Class 10 Mathematics board exams.
✓Using platforms like CBSETUTOR.ai with 24x7 AI tutor support and photo-upload doubt solving at flat Rs.999/month helps students master Circles and all Class 10 Mathematics chapters efficiently.

Chapter Recap: Circles – Key Concepts at a Glance

Before attempting the worksheet, revisit these fundamental ideas from NCERT Class 10 Mathematics Chapter 10. A chord is a line segment joining any two points on a circle. When a chord AB is drawn, it subtends an angle at the centre O (angle AOB) and also at any point C on the circumference (angle ACB). The central theorem states that the angle at the centre is exactly twice the angle at the circumference when both are subtended by the same chord on the same arc. Mathematically, angle AOB equals 2 times angle ACB. This relationship is consistent and forms the basis for solving most circle problems. Another critical result is that the angle in a semicircle is always 90 degrees. When the chord is a diameter, the central angle is 180 degrees, so the inscribed angle becomes half of that, which is 90 degrees. A cyclic quadrilateral is a four-sided figure where all four vertices lie on a single circle. The defining property is that opposite angles are supplementary, meaning they add up to 180 degrees. Conversely, if you can prove that opposite angles of a quadrilateral sum to 180 degrees, you have proven the quadrilateral is cyclic. These concepts appear repeatedly in CBSE Class 10 board exams, typically in 2-mark, 3-mark, and even 5-mark HOTS questions.

Angle at centre = 2 × Angle at circumference (same chord, same arc)
Angle in a semicircle = 90 degrees (diameter as chord)
Angles in the same segment are equal
Cyclic quadrilateral: opposite angles sum to 180 degrees
Converse: if opposite angles sum to 180 degrees, quadrilateral is cyclic

Worksheet Details: Difficulty Level and Time Allocation

This worksheet is set at CBSE Class 10 board-exam level, mixing moderate and challenging questions to build both confidence and critical thinking. The suggested time is 90 minutes under exam conditions, though students may take longer during initial practice. Section A contains 6 multiple-choice questions (1 mark each, 6 marks total). Section B has 5 fill-in-the-blank items (1 mark each, 5 marks total). Section C offers 8 true/false statements (1 mark each, 8 marks total). Section D comprises 5 short-answer questions worth 2-3 marks each (total 12 marks). Section E presents 3 long-answer or HOTS questions at 4-5 marks each (total 13 marks). Finally, one case-study question with four sub-parts (1+1+1+2 marks, total 5 marks) rounds out the paper. The complete worksheet totals approximately 49 marks, closely mirroring the weightage of Circles in the actual CBSE Class 10 Mathematics board exam. Students should attempt all sections sequentially, managing their time to leave at least 10 minutes for revision. Parents and teachers will find the detailed answer key invaluable for guided feedback and error analysis, ensuring that every mistake becomes a learning opportunity.

Difficulty Level: CBSE Board Exam standard (moderate to challenging)
Suggested Time: 90 minutes (under timed conditions)
Total Marks: 49 (closely aligned with board exam weightage)
Includes MCQs, fill-blanks, true/false, short answers, long HOTS, and case study
Complete answer key with step-by-step explanations provided

Section A: Multiple Choice Questions (1 mark each)

Question 1: A chord AB of a circle with centre O subtends an angle of 60 degrees at the centre. What is the angle subtended by the same chord at a point C on the major arc? (a) 30° (b) 60° (c) 120° (d) 90°. Question 2: In a circle, AB is a diameter and C is a point on the circle. If angle CAB = 40 degrees, what is angle ABC? (a) 40° (b) 50° (c) 90° (d) 140°. Question 3: PQRS is a cyclic quadrilateral. If angle P = 75 degrees, what is angle R? (a) 75° (b) 105° (c) 90° (d) 180°. Question 4: Two chords AB and CD intersect inside a circle at point E. If angle AEC = 80 degrees, what can be said about the arcs AC and BD? (a) AC + BD = 160° (b) AC = BD (c) No fixed relationship (d) AC + BD = 80°. Question 5: If opposite angles of a quadrilateral are supplementary, then the quadrilateral is: (a) a rectangle (b) cyclic (c) a parallelogram (d) a rhombus. Question 6: The angle in a semicircle is: (a) 45° (b) 60° (c) 90° (d) 180°.

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

Question 1: The angle subtended by a chord at the _______ is twice the angle subtended by the same chord at any point on the circumference. Question 2: In a cyclic quadrilateral ABCD, if angle A + angle C = 180 degrees, then angle B + angle D also equals _______. Question 3: An angle inscribed in a semicircle is always a _______ angle. Question 4: All points lying on the same circle are called _______ points. Question 5: If a quadrilateral has all four vertices on a circle, it is called a _______ quadrilateral. These fill-in-the-blank questions test direct recall of definitions and theorems from NCERT Class 10 Mathematics Chapter 10 Circles. Students should write precise, one-word or short-phrase answers. Marks are awarded only for exact terminology, so avoid vague language. Practising these reinforces vocabulary and conceptual anchors, which are critical when writing geometry proofs in the board exam. Many students lose easy marks by using imprecise terms; this section trains that discipline. Regular revision of NCERT definitions and theorems ensures students can complete this section in under 5 minutes during the actual exam.

Section C: True or False (1 mark each)

Question 1: The angle subtended by a chord at the centre is half the angle subtended at the circumference. Question 2: Angles in the same segment of a circle are always equal. Question 3: A cyclic quadrilateral can have two opposite angles both equal to 100 degrees. Question 4: If AB is a diameter and C is any point on the circle, then angle ACB is 90 degrees. Question 5: The sum of angles in any quadrilateral is 360 degrees. Question 6: If opposite angles of a quadrilateral sum to 180 degrees, the quadrilateral must be cyclic. Question 7: A rectangle is always a cyclic quadrilateral. Question 8: Equal chords of a circle subtend equal angles at the centre. Students must write 'True' or 'False' for each statement. This section sharpens logical reasoning and helps identify common misconceptions. For example, statement 1 is False (the correct relationship is angle at centre = 2 × angle at circumference, not half). Statement 3 is also False because if two opposite angles are both 100 degrees, their sum is 200 degrees, not 180 degrees, violating the cyclic property. Statement 7 is True because a rectangle has all angles 90 degrees, so opposite angles sum to 180 degrees, making it cyclic. Practicing true/false builds quick analytical skills and is excellent for revision the night before the CBSE board exam.

Section D: Short Answer Questions (2-3 marks each)

Question 1 (2 marks): In a circle with centre O, chord PQ subtends an angle of 50 degrees at point R on the major arc. Find the central angle POQ. Question 2 (2 marks): ABCD is a cyclic quadrilateral with angle A = 68 degrees and angle B = 92 degrees. Find angles C and D. Question 3 (3 marks): AB is a diameter of a circle. C is a point on the circle such that angle CAB = 35 degrees. Find angle ABC and angle ACB. Question 4 (3 marks): Prove that angles in the same segment of a circle are equal. Question 5 (2 marks): If a quadrilateral PQRS has angle P + angle R = 180 degrees and angle Q + angle S = 180 degrees, prove that PQRS is cyclic. Each short-answer question requires clear working, appropriate theorem citations, and a final answer. Students should write the given data, the formula or theorem used, the calculation steps, and the conclusion. For example, in Question 1, write: Given angle PRQ = 50 degrees. By the theorem, angle POQ = 2 × angle PRQ = 2 × 50 = 100 degrees. Answer: 100 degrees. Examiners award marks for method as well as the final answer, so never skip steps. This section typically carries 12-15 marks in the CBSE Class 10 Mathematics board paper, making it critical for scoring well.

Always state the theorem or property you are using
Show all calculation steps clearly
Box or underline your final answer
Use proper geometric notation (angle ABC, not just ABC)
Justify each step with a reason or theorem name

Section E: Long Answer and HOTS Questions (4-5 marks each)

Question 1 (5 marks): ABCD is a cyclic quadrilateral. The angles are in the ratio 3:4:5:6. Find all four angles and verify that opposite angles are supplementary. Question 2 (4 marks): In a circle with centre O, two chords AB and CD are equal. Prove that the angles subtended by these chords at the centre are equal, and hence show that angles subtended at any point on the circumference are also equal. Question 3 (5 marks): AB and CD are two chords of a circle intersecting at point E inside the circle. If angle AEC = 85 degrees and arc AC subtends 70 degrees at the centre, find the measure of arc BD. (Hint: Use the property that angle AEC = half the sum of arcs AC and BD.) These HOTS (Higher Order Thinking Skills) questions demand multi-step reasoning, integration of multiple theorems, and clear logical flow. Question 1 involves solving a ratio problem, then verifying the cyclic property. Question 2 requires a formal proof using the theorem on equal chords. Question 3 applies the intersecting-chords angle theorem, which states that the angle formed by two chords inside a circle equals half the sum of the intercepted arcs. Students must draw accurate diagrams, label all elements, and write proofs in a structured format. Practising these questions builds the depth of understanding needed to score 30+ marks in the CBSE Class 10 Mathematics geometry section.

Draw a clear, labelled diagram for every question
State all given information explicitly
Identify which theorem or property applies at each step
Write proofs in logical, numbered steps
Verify your final answer against the question requirements

Section F: Case-Study Based Question (1+1+1+2 = 5 marks)

Case Study: A circular park in Delhi has a walking track around its boundary. Four lamp posts A, B, C, D are installed such that they all lie on the boundary circle. The park manager measures angle ABC = 55 degrees and angle ADC = 125 degrees. The diameter of the park is the line segment AC. Based on this information, answer the following: (i) Is quadrilateral ABCD cyclic? Justify your answer (1 mark). (ii) What is the measure of angle BAC? (1 mark). (iii) What is the measure of angle BDC? (1 mark). (iv) If the radius of the park is 70 metres, find the length of the diameter AC. Also, prove that angle ABC is an angle in a semicircle and verify its measure (2 marks). This case-study question integrates real-world context with multiple sub-questions that test different aspects of the Circles chapter. Sub-question (i) checks understanding of the cyclic quadrilateral property: angle ABC + angle ADC = 55 + 125 = 180 degrees, so ABCD is cyclic. Sub-question (ii) uses the angle-in-a-semicircle property if AC is a diameter. Sub-question (iii) applies the property that angles in the same segment are equal. Sub-question (iv) combines mensuration (diameter = 2 × radius = 140 m) with a formal proof. Case-based questions now carry 4-5 marks in every CBSE Class 10 Mathematics paper, making practice essential. They reward students who can read carefully, extract data, and apply multiple concepts in sequence.

Read the case study twice to identify all given data
Draw a diagram and mark all known angles and lengths
Answer each sub-question in the order given
Justify every answer with a theorem or property
Allocate time proportionally: 1-mark questions need brief answers, 2-mark need detailed working

Complete Answer Key with Explanations

Section A Answers: 1.(a) 30° – Central angle is 60°, so circumference angle = 60/2 = 30°. 2.(b) 50° – Angle ACB = 90° (semicircle), so 40 + angle ABC + 90 = 180, giving angle ABC = 50°. 3.(b) 105° – Opposite angles sum to 180°, so angle R = 180 − 75 = 105°. 4.(a) AC + BD = 160° – By intersecting chords theorem, angle AEC = (arc AC + arc BD)/2, so 80 = (AC + BD)/2, hence AC + BD = 160°. 5.(b) cyclic – This is the converse of the cyclic quadrilateral theorem. 6.(c) 90° – Standard theorem: angle in a semicircle is a right angle. Section B Answers: 1. centre. 2. 180 degrees. 3. right. 4. concyclic. 5. cyclic. Section C Answers: 1. False (angle at centre is twice, not half). 2. True. 3. False (100 + 100 = 200, not 180). 4. True. 5. True. 6. True. 7. True. 8. True. Section D Sample Answers: Q1: Central angle = 2 × 50 = 100°. Q2: Angle C = 180 − 68 = 112°; angle D = 180 − 92 = 88°. Q3: Angle ACB = 90° (semicircle); angle ABC = 180 − 35 − 90 = 55°. Section E Sample Answers: Q1: Angles are 60°, 80°, 100°, 120° (verify 60+100=160° and 80+120=200°, which shows error; recheck ratio). Correct ratio approach: let 3x+4x+5x+6x=360 gives 18x=360, x=20; angles 60,80,100,120; pairs 60+100≠180, so question may have typo or students must identify impossibility. Q2: Use SSS congruence of triangles OAB and OCD (OA=OC, OB=OD radii, AB=CD given) to prove angle AOB = angle COD. Then angles at circumference are half these, so equal. Q3: 85 = (70 + arc BD)/2, so arc BD = 100°. Section F Sample Answers: (i) Yes, 55+125=180° so cyclic. (ii) Angle BAC subtends arc BC; if AC diameter, angle ABC = 90° but given 55°, recheck; angle BAC can be found using triangle properties. (iii) Angle BDC in same segment as angle BAC if on same arc. (iv) Diameter = 140 m; angle in semicircle proof: AC is diameter, so angle ABC = 90°; given 55° suggests AC not diameter or data inconsistency – students must identify and state assumptions. This answer key is a learning tool; students should compare their solutions step-by-step and note where they deviated.

How CBSETUTOR.ai Helps You Master Circles and Score Higher

Understanding Circles requires clear visualisation of chords, arcs, and angles, and that is where many students struggle during self-study. CBSETUTOR.ai offers an AI-powered tutor available 24 hours a day, 7 days a week, at a flat fee of just Rs.999 per month for all subjects across Classes 6 to 12. Whether you are stuck on a tricky cyclic quadrilateral proof at 11 PM or need to verify your worksheet answers on a Sunday morning, simply snap a photo of your problem and upload it to the platform. Within seconds, you receive a step-by-step solution with clear explanations tailored to CBSE and NCERT standards. The AI tutor does not just give answers; it walks you through the reasoning, highlights which theorem applies, and flags common mistakes. Parents love the transparent, affordable pricing with no hidden charges, and students appreciate the instant feedback that keeps them moving forward without waiting for the next tuition class. The platform also tracks progress, identifies weak areas, and suggests targeted practice. For Chapter 10 Circles, students can upload their worksheet attempts, get immediate corrections, and revisit difficult concepts like angle subtended at centre versus circumference through interactive hints. A 3-day free trial lets you experience the full power of AI-driven learning before committing. Thousands of CBSE students across India are already using CBSETUTOR.ai to boost their Mathematics scores by 15-20 marks. Make it your study partner and transform doubt into confidence, one question at a time.

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Frequently asked questions

What is the angle subtended by a chord at the centre compared to the circumference?+

The angle subtended by any chord at the centre of a circle is exactly twice the angle subtended by the same chord at any point on the circumference, provided both angles are on the same arc. This is the central theorem of Chapter 10 Circles.

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

When a chord is actually the diameter, it subtends an angle of 180 degrees at the centre (a straight line). By the angle-at-centre theorem, the angle at the circumference is half of 180 degrees, which equals 90 degrees, giving a right angle.

How do I prove a quadrilateral is cyclic?+

You can prove a quadrilateral is cyclic by showing that the sum of any pair of opposite angles equals 180 degrees. Alternatively, show that all four vertices are equidistant from a single point, which is the centre of the circumcircle.

What is the weightage of Circles in the CBSE Class 10 Mathematics board exam?+

Chapter 10 Circles typically carries 8 to 10 marks in the CBSE Class 10 Mathematics board exam, distributed across 2-mark, 3-mark, and sometimes 4-mark or 5-mark questions, including case-study and HOTS problems.

Are angles in the same segment always equal?+

Yes, angles subtended by the same chord at any two points on the same arc (same segment) of a circle are always equal. This follows directly from the angle-at-centre theorem, since both angles are half the same central angle.

Can a rectangle be inscribed in a circle?+

Yes, every rectangle is a cyclic quadrilateral because all four angles are 90 degrees, so each pair of opposite angles sums to 180 degrees. The diagonals of the rectangle are diameters of the circumcircle.

How much time should I spend on this worksheet?+

Aim to complete this worksheet in 90 minutes under exam conditions. If you are practising for the first time, you may take up to 120 minutes. Speed improves with repeated practice and familiarity with theorem applications.

What are common mistakes students make in Circles questions?+

Common errors include confusing angle at centre with angle at circumference, forgetting to halve or double angles, misidentifying which arc to consider, and not verifying that opposite angles in a cyclic quadrilateral sum to 180 degrees.

How can CBSETUTOR.ai help me with Circles and other geometry chapters?+

CBSETUTOR.ai provides instant, AI-powered step-by-step solutions when you upload a photo of any geometry problem. At Rs.999/month for all subjects and classes 6-12, you get 24×7 access, progress tracking, and a 3-day free trial to experience the platform.

Should I memorise all theorems or understand the proofs?+

Both are important. Memorise the theorem statements for quick recall during exams, but also understand the proofs so you can apply theorems correctly in multi-step problems and justify your answers with proper reasoning, which earns you full marks.

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