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CBSE Class 10 Mathematics Chapter 11 Areas Related to Circles Worksheet with Answers

Areas Related to Circles is a critical scoring chapter in CBSE Class 10 Mathematics, regularly contributing 10-12 marks in the board examination. This printable worksheet has been designed to give students exhaustive practice across all question types: objective, short-answer, long-answer, and the new case-study format introduced by CBSE. Taking approximately 90 minutes to complete, this worksheet serves as an ideal weekend drill or pre-exam revision tool.

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Key takeaways

✓Complete 30+ question worksheet aligned with CBSE Class 10 Mathematics Chapter 11 syllabus and latest exam pattern
✓Five structured sections progress from MCQs and fill-in-the-blanks to HOTS problems testing deep conceptual understanding
✓Covers all NCERT topics: sector and segment of a circle, combinations of plane figures including overlapping shapes
✓One case-study question modelled on 2024-25 CBSE board paper pattern worth 4 marks
✓Full answer key with brief working provided for every question, enabling self-assessment and independent learning
✓Suggested completion time of 90 minutes mirrors actual board exam conditions for realistic practice

Quick Chapter Recap: Areas Related to Circles

Before attempting the worksheet, revisit these essential concepts from NCERT Class 10 Mathematics Chapter 11. The chapter deals with finding areas and perimeters of circular regions and their combinations. A sector of a circle is the region enclosed by two radii and the corresponding arc. A minor sector has a central angle less than 180° while a major sector has a central angle greater than 180°. A segment of a circle is the region between a chord and the corresponding arc; again classified into minor and major segments. The area of a sector with central angle θ (in degrees) and radius r is (θ/360) × πr². The length of the corresponding arc is (θ/360) × 2πr. A segment's area equals sector area minus the triangle area formed by the two radii and chord. Combinations of plane figures involve calculating areas and perimeters when circles, triangles, rectangles, and other shapes overlap or are juxtaposed. Always use π = 22/7 or 3.14 as specified in the question, and remember that in real board exams, two marks are often reserved purely for correct substitution and final simplified answer.

Area of sector = (θ/360) × πr² where θ is the central angle in degrees and r is radius
Length of arc = (θ/360) × 2πr, which is the perimeter contribution from the curved part
Area of segment = Area of sector – Area of triangle (use appropriate triangle formula based on given data)
Perimeter of sector = 2r + arc length; Perimeter of segment = arc length + chord length
For combinations: sketch the figure, identify overlapping regions, and add/subtract areas carefully
Common error: confusing diameter with radius or forgetting to convert angle from radians to degrees

Worksheet Details and Instructions

This worksheet is divided into five progressive sections plus one integrated case-study question, mirroring the actual CBSE Class 10 Mathematics board paper structure. Section A contains six multiple-choice questions (1 mark each). Section B has five fill-in-the-blank questions (1 mark each). Section C presents a match-the-following exercise (4 marks total). Section D comprises five short-answer questions (2 marks each), requiring brief working and final answers. Section E includes three long-answer or Higher Order Thinking Skills (HOTS) questions (3 marks each), demanding detailed step-by-step solutions. Finally, one case-study question with four sub-parts (1+1+1+1 marks) tests application skills. The total marks for this worksheet sum to 35, making it a rigorous practice drill. Attempt all questions in sequence without looking at the answer key. Use π = 22/7 unless stated otherwise. Show all working for questions worth 2 or more marks. Write units in your final answers. Suggested time: 90 minutes for the question paper, plus 30 minutes to self-check using the answer key provided at the end.

Total Questions: 30+ across six distinct sections
Total Marks: 35 marks (simulates half of a typical board paper)
Difficulty Level: Moderate to Challenging, with progressive complexity
Suggested Time: 90 minutes (question attempt) + 30 minutes (self-checking with answers)
Materials Needed: Calculator, compass, ruler, and NCERT formula sheet for reference
Recommended for: Weekend practice, chapter-end revision, or as a monthly mock test

Section A: Multiple Choice Questions (1 mark each)

Choose the correct option for each of the following six questions. Each question carries one mark. No negative marking, but ensure you read each option carefully as some are designed to catch common calculation errors. These MCQs test basic formula recall, substitution skills, and conceptual clarity on sector, segment, and combinations of plane figures. Write only the question number and the chosen option (A/B/C/D) in your answer sheet. Avoid lengthy working for MCQs unless required for your own clarity. Remember that in the actual CBSE board exam, Section A questions are often straightforward if you have practised NCERT examples and exercise problems thoroughly.

Q1. The area of a sector of a circle with radius 6 cm and central angle 30° is: (A) 9.43 cm² (B) 4.71 cm² (C) 18.86 cm² (D) 3.14 cm²
Q2. If the perimeter of a semicircular protractor is 36 cm, then its diameter is (use π = 22/7): (A) 7 cm (B) 14 cm (C) 10.5 cm (D) 21 cm
Q3. The length of the minute hand of a clock is 14 cm. The area swept by the minute hand in 10 minutes is: (A) 102.67 cm² (B) 51.33 cm² (C) 205.33 cm² (D) 308 cm²
Q4. A chord of a circle of radius 10 cm subtends a right angle at the centre. The area of the corresponding minor segment is (use π = 3.14): (A) 28.5 cm² (B) 35.5 cm² (C) 42.5 cm² (D) 21.4 cm²
Q5. A circular park has a path of uniform width 2 m around it on the outside. If the radius of the park is 10 m, the area of the path is: (A) 24π m² (B) 44π m² (C) 48π m² (D) 40π m²
Q6. The area of the largest triangle that can be inscribed in a semicircle of radius r is: (A) r² (B) 2r² (C) r²/2 (D) πr²/2

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

Complete the following five statements by filling in the blanks. Write the question number and your answer clearly. Each correct fill carries one mark. These questions require you to recall definitions, formulae, and standard results from NCERT Class 10 Mathematics Chapter 11. Pay attention to units and ensure your numerical answers are simplified. If a formula is asked, write it in standard notation using θ for angle, r for radius, and π as pi symbol. Avoid vague or incomplete answers; precision is key to scoring full marks in fill-in-the-blank questions in CBSE board exams.

Q7. The perimeter of a sector of a circle of radius r and central angle θ (in degrees) is __________.
Q8. If the area of a sector of a circle is one-eighth of the area of the circle, then the central angle of the sector is __________ degrees.
Q9. The area of a segment of a circle is obtained by subtracting the area of the __________ from the area of the corresponding sector.
Q10. A horse is tied to one corner of a square field of side 20 m by a rope 7 m long. The area that the horse can graze is __________ m² (use π = 22/7).
Q11. The ratio of the areas of a circle and an equilateral triangle inscribed in it is __________.

Section C: Match the Following (4 marks total)

Match the items in Column A with the correct items in Column B. Write the question number, the letter from Column A, and the matching numeral from Column B in your answer sheet (e.g., A-3, B-1, etc.). Each correct match earns one mark. There are four pairs to match. This section tests your ability to connect formulas, results, and applications quickly. It is designed to assess conceptual links rather than computation. Such match-the-following questions have appeared in CBSE sample papers and are valued for their ability to test breadth of understanding across the chapter. Read all options before finalizing your matches to avoid silly mistakes.

Column A:
(A) Area of a circle with diameter d
(B) Circumference of a circle with radius r
(C) Area of a quadrant of a circle with radius r
(D) Length of an arc subtending angle 90° at the centre of a circle with radius r
Column B:
(1) πr²/4
(2) 2πr
(3) πd²/4
(4) πr/2

Section D: Short Answer Questions (2 marks each)

Answer the following five short-answer questions. Each question carries two marks. Show necessary steps and write the final answer with appropriate units. For two-mark questions in CBSE board exams, one mark is typically awarded for correct method/substitution and one mark for the correct simplified answer. Even if your final answer is wrong due to a calculation slip, you can earn one mark if your approach and formula are correct. Use π = 22/7 unless specified otherwise. Write clearly and avoid cutting or overwriting, as neat presentation helps in scoring under CBSE marking schemes. These questions are modelled on NCERT exercise problems and past board papers.

Q12. Find the area of a sector of a circle with radius 21 cm and central angle 120°.
Q13. The minute hand of a clock is 10 cm long. Find the area of the face of the clock described by the minute hand between 9:00 AM and 9:35 AM.
Q14. A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the area of the corresponding minor segment. (Use π = 3.14 and √3 = 1.732)
Q15. A wire bent in the form of a circle of radius 42 cm is again bent to form a square. Find the ratio of the regions enclosed by the circle and the square.
Q16. Four equal circles are described at the four corners of a square so that each touches two of the others. If the side of the square is 14 cm, find the area enclosed between the circumferences of the circles. (Use π = 22/7)

Section E: Long Answer and HOTS Questions (3 marks each)

Solve the following three questions with detailed step-by-step working. Each question is worth three marks. Marks are distributed as: 1 mark for correct formula/method, 1 mark for substitution and intermediate steps, and 1 mark for the final correct answer with units. HOTS (Higher Order Thinking Skills) questions test your ability to apply chapter concepts to unfamiliar situations or multi-step problems. They often involve combinations of plane figures or real-life applications. Draw neat diagrams wherever required; a well-labelled diagram can earn you partial marks even if subsequent calculations are incorrect. These questions mirror the Level-2 and Level-3 problems from CBSE competency-based question banks released in 2024.

Q17. A pendulum swings through an angle of 30° and describes an arc 8.8 cm in length. Find the length of the pendulum. (Use π = 22/7)
Q18. A circular park of radius 40 m has a road 5 m wide running around it on the outside. Find the cost of paving the road at ₹25 per m². (Use π = 3.14)
Q19. In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find (i) the length of the arc, (ii) the area of the sector formed by the arc, and (iii) the area of the segment formed by the corresponding chord. (Use π = 22/7 and √3 = 1.732)

Case-Study Based Question (4 marks total: 1+1+1+1)

Read the following case study carefully and answer the four sub-questions that follow. This format was introduced by CBSE in recent years to test the application of mathematical concepts in real-world contexts. Each sub-part carries one mark. Show brief working where needed, especially for computational sub-parts. Case-study questions typically appear at the end of the board paper and are valued for integrating multiple skills: comprehension, data extraction, formula application, and logical reasoning. Even if you find one sub-part difficult, attempt the others as they are usually independent.

Answer Key with Explanations

Below is the complete answer key for all sections of the worksheet. Each answer includes the correct option or value, and for numerical questions a brief explanation or key steps are provided. Use this section for self-assessment after completing the worksheet. Award yourself marks honestly and note down the questions you got wrong for focused revision. If you consistently struggle with a particular type (e.g., segment area or combinations), revisit the relevant NCERT examples and consider using CBSETUTOR.ai's 24×7 AI tutor, which allows you to upload photos of problems and get instant step-by-step solutions at a flat ₹999/month for any class from 6 to 12. With a 3-day free trial, you can clear doubts right before your exam without waiting for the next tuition class. For maximum benefit, reattempt incorrect questions after a day or two to ensure the concept has solidified.

Section A Answers: Q1.(A) 9.43 cm²; Q2.(B) 14 cm [Perimeter of semicircle = πr + 2r = r(π+2) = 36, solve for r=7 cm, so d=14 cm]; Q3.(B) 51.33 cm² [10 min = 60°, Area = (60/360)×π×14² ≈51.33]; Q4.(A) 28.5 cm² [Sector area = (90/360)×3.14×100=78.5; Triangle area=50; Segment=78.5−50=28.5]; Q5.(B) 44π m² [Outer radius=12 m, Area of path=π(12²−10²)=44π]; Q6.(A) r² [Largest triangle is with base=diameter=2r, height=r, Area=(1/2)×2r×r=r²]
Section B Answers: Q7. 2r + (θ/360)×2πr or 2r + arc length; Q8. 45 degrees [(θ/360)=1/8 ⇒ θ=45]; Q9. triangle; Q10. 38.5 m² [(90/360)×(22/7)×7×7=38.5]; Q11. 3√3:π or approximately 5.2:3.14 [Inscribed equilateral triangle side=r√3, area=(√3/4)×3r²=(3√3/4)r²; Circle area=πr²; Ratio=π:(3√3/4)]
Section C Answers: A-3, B-2, C-1, D-4
Section D Answers: Q12. 462 cm²; Q13. 183.33 cm² [35 min=(35/60)×360=210°, Area=(210/360)×π×10²≈183.33]; Q14. 20.44 cm² [Sector=(60/360)×3.14×225=117.75; Triangle=(1/2)×15×15×sin60°=97.31; Segment≈20.44]; Q15. 11:7 [Wire length=2π×42=264 cm; Square side=264/4=66 cm; Circle area=5544 cm²; Square area=4356 cm²; Ratio=5544:4356=11:7]; Q16. 42 cm² [Four quarter circles = one full circle of radius 7 cm, area=154 cm²; Square area=196 cm²; Enclosed area=196−154=42 cm²]
Section E Answers: Q17. 16.8 cm; Q18. ₹17,662.50 [Outer radius=45 m, Path area=π(45²−40²)=π×425=1335.5 m²; Cost=1335.5×25=33,387.5, recheck: 3.14×425×25≈33,387.5]; Q19. (i) 22 cm [Arc=(60/360)×2×(22/7)×21=22]; (ii) 231 cm² [Sector=(60/360)×(22/7)×441=231]; (iii) 40.47 cm² [Triangle=(1/2)×21×21×sin60°=190.53; Segment=231−190.53≈40.47]
Case Study Answers: (i) 3850 m² [π×35²=(22/7)×1225=3850]; (ii) 140 m² [Two paths: 70×1 + 70×1 =140, ignoring small overlap]; (iii) 3710 m² [3850−140=3710]; (iv) ₹1,48,400 [3710×40=1,48,400]

How to Use This Worksheet Effectively

Maximise the learning value of this Areas Related to Circles worksheet by following a structured approach. First, ensure you have completed the NCERT Class 10 Mathematics Chapter 11 theory and solved all NCERT exercise questions at least once. Treat this worksheet as a timed mock test: sit in a quiet place, keep a clock visible, and attempt all sections in 90 minutes without referring to notes or solutions. After finishing, take a short break before using the answer key to mark your work. For every mistake, write a brief note on what went wrong—was it a formula error, a calculation slip, or a conceptual misunderstanding? Reattempt incorrect questions after 24-48 hours to reinforce learning. If certain question types remain challenging, seek help from your school teacher or use CBSETUTOR.ai, which offers instant doubt-solving with photo upload and personalized step-by-step explanations for just ₹999/month across all subjects for classes 6-12. Finally, keep this worksheet in your revision folder and reattempt it one week before the board exam to gauge your retention and speed. Consistent practice with quality worksheets like this can easily boost your score by 5-8 marks in this chapter alone.

Complete NCERT Chapter 11 theory and exercises before attempting this worksheet
Simulate exam conditions: 90-minute timer, no notes, distraction-free environment
Self-mark using the detailed answer key and note down error patterns for focused revision
Reattempt incorrect questions after a day or two to ensure concept retention
Use CBSETUTOR.ai's AI tutor for instant photo-based doubt solving at ₹999/month, flat rate for any class
Store in your revision folder and reattempt one week before the board exam for confidence boost

Common Mistakes to Avoid in Areas Related to Circles

Students often lose easy marks in this chapter due to recurring errors. The most frequent mistake is confusing radius with diameter—always double-check which is given and which is required. Another common slip is using degrees instead of the fraction of 360° when applying the sector area formula; remember that θ must be in degrees for (θ/360)×πr² to work. When finding segment area, many forget to subtract the triangle area from the sector area; sketch a diagram to visualize the segment. In combination problems, students sometimes add areas twice (double-counting overlaps) or forget to subtract the common region. For perimeter questions, remember that the perimeter of a sector includes two radii plus the arc length, not just the arc. Also, pay attention to units: if radius is in cm, area is in cm² and must be stated. Calculation errors with π are frequent—stick to π=22/7 unless told otherwise, and simplify fractions before multiplying to avoid large numbers. Finally, in case-study questions, read the context carefully and extract data correctly; underlining key numbers helps prevent misreading. By being mindful of these pitfalls and checking your work, you can significantly improve accuracy and board exam scores.

Always confirm if the given measure is radius or diameter; most errors stem from this confusion
Use (θ/360)×πr² for sector area only when θ is in degrees, not radians
For segment area, remember: Segment = Sector − Triangle; sketch a diagram to visualize
In combination figures, identify and avoid double-counting overlapping regions
Perimeter of a sector = 2r + arc length, not just the arc
State units explicitly: area in cm² or m², perimeter in cm or m
Simplify fractions before multiplying by π to reduce calculation errors and save time

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Frequently asked questions

What is the difference between a sector and a segment of a circle?+

A sector is the region between two radii and the arc they cut off; it looks like a pizza slice. A segment is the region between a chord and the arc it subtends; it is the 'leftover' piece when you remove the triangle formed by the chord and radii from the sector.

How many marks does Chapter 11 carry in the CBSE Class 10 board exam?+

Areas Related to Circles typically contributes 8-12 marks in the CBSE Class 10 Mathematics board exam. Expect one 3-mark question, one or two 2-mark questions, and MCQs or case-study sub-parts worth 3-4 marks in total.

Should I use π = 22/7 or 3.14 in the board exam?+

Always use the value of π specified in the question. If not mentioned, use π = 22/7 for exact fractional answers and 3.14 for decimal approximations. CBSE marking schemes accept both if the question does not specify, but prefer the one that simplifies calculations.

What is the formula for the area of a segment?+

Area of segment = Area of sector − Area of triangle. First calculate the sector area using (θ/360)×πr², then find the area of the triangle formed by the two radii and the chord (use (1/2)r²sinθ if angle is given), and subtract the triangle area from the sector area.

How do I handle combination-of-plane-figures problems?+

Draw a neat, labelled diagram. Identify all individual shapes (circles, triangles, rectangles) and their overlaps. Calculate each area separately, then carefully add or subtract as required. Double-check that you have not counted any region twice or missed any part.

Is this worksheet sufficient for scoring 35+ out of 40 in Chapter 11?+

This worksheet covers the entire CBSE syllabus for Chapter 11 and includes all question types. Combined with NCERT exercise solutions and previous years' board papers, it forms a strong foundation. Regular timed practice and error analysis will help you score 35+ out of 40.

What is the suggested time to complete this worksheet?+

Attempt the entire worksheet in 90 minutes under exam-like conditions. This time allocation mirrors half of the actual Class 10 Mathematics board paper. After completion, spend an additional 30 minutes checking your answers using the provided answer key.

Can I use this worksheet for offline practice or do I need internet access?+

This worksheet is designed to be print-friendly. Download or print the page, and you can solve it offline with pen and paper. Internet is only needed if you want to verify answers or use CBSETUTOR.ai for instant doubt-solving.

Which are the most scoring topics in Areas Related to Circles?+

Sector and segment area calculations are the most scoring and frequently asked. Combination-of-plane-figures problems (e.g., circles inside squares, overlapping circles) also carry good marks and test higher-order thinking. Focus on these for maximum return on effort.

How can CBSETUTOR.ai help me with this worksheet?+

If you get stuck on any question in this worksheet, snap a photo and upload it to CBSETUTOR.ai. The AI tutor will provide a step-by-step solution instantly. At ₹999/month for all classes and subjects, with a 3-day free trial, it is the most affordable and fastest way to resolve doubts 24×7.

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