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CBSE Class 9 Mathematics Chapter 11 Surface Areas and Volumes Worksheet with Answers

Surface Areas and Volumes is a scoring chapter in CBSE Class 9 Mathematics that directly builds foundations for Class 10 board exams. This worksheet offers focused practice on all formula-based problems, from basic surface area calculations to complex volume conversions. Designed for 90 minutes of concentrated work, it mirrors the CBSE exam pattern with MCQs, short-answer, long-answer, and case-study questions.

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Key takeaways

✓Printable 90-minute worksheet covering all question types from CBSE Class 9 Mathematics Chapter 11 Surface Areas and Volumes.
✓Six multiple-choice questions test conceptual understanding of formulae for cubes, cuboids, cylinders, cones, spheres and hemispheres.
✓Five short-answer questions provide practice calculating surface areas and volumes of common 3D shapes.
✓Three HOTS long-answer questions challenge students with multi-step problems involving combined solids and real-world applications.
✓Complete answer key with step-by-step explanations helps students self-assess and learn from mistakes.
✓One case-study question mirrors the latest CBSE exam pattern introduced in 2024-25 term exams.
✓Difficulty level: Moderate to Challenging—suitable for thorough revision and competitive exam preparation.

Chapter 11 Quick Recap: Surface Areas and Volumes

Chapter 11 introduces mensuration of three-dimensional solids—cubes, cuboids, cylinders, cones, spheres, and hemispheres. Students must memorise and apply formulae for total surface area, lateral (curved) surface area, and volume of each shape. A cube with side 'a' has surface area 6a² and volume a³. A cuboid with length l, breadth b, and height h has total surface area 2(lb + bh + hl) and volume lbh. For a cylinder of radius r and height h, curved surface area is 2πrh, total surface area is 2πr(r + h), and volume is πr²h. A cone with radius r, height h, and slant height l = √(r² + h²) has curved surface area πrl, total surface area πr(r + l), and volume ⅓πr²h. A sphere of radius r has surface area 4πr² and volume ⅘πr³, while a hemisphere has curved surface area 2πr², total surface area 3πr², and volume ⅔πr³. The 2025 CBSE paper often includes 3–4 questions worth 9–12 marks from this chapter.

Cube: TSA = 6a², Volume = a³
Cuboid: TSA = 2(lb + bh + hl), Volume = lbh
Cylinder: CSA = 2πrh, TSA = 2πr(r+h), Volume = πr²h
Cone: CSA = πrl, TSA = πr(r+l), Volume = ⅓πr²h
Sphere: SA = 4πr², Volume = ⅘πr³
Hemisphere: CSA = 2πr², TSA = 3πr², Volume = ⅔πr³

Worksheet Instructions and Difficulty Level

This worksheet is designed for moderate to challenging practice and should be completed in 90 minutes under exam-like conditions. Students are advised to keep a scientific calculator, NCERT textbook, and rough sheets handy. Each section tests different cognitive skills: Section A (MCQs) tests quick recall and formula application; Section B (fill-in-the-blanks) reinforces direct formula usage; Section C (matching or true/false) checks conceptual clarity; Section D (short-answer) requires two- to three-step calculations; Section E (long-answer) demands multi-step problem solving and logical reasoning. The case-study question integrates real-world scenarios with mathematical modelling. After completing the worksheet, students should review the detailed answer key, paying special attention to unit conversions and common mistakes such as confusing radius with diameter or using slant height instead of vertical height. This worksheet aligns with the latest CBSE syllabus for 2024-25 and includes question formats seen in recent board exams.

Total time: 90 minutes
Difficulty: Moderate to Challenging
Materials needed: calculator, compass, ruler, NCERT textbook
Recommended for: pre-exam revision, weekend practice, self-assessment
Marking scheme mirrors CBSE board pattern

Section A: Multiple Choice Questions (1 mark each)

This section contains six multiple-choice questions testing your understanding of basic formulae and their applications. Each question carries one mark. Choose the correct option and write it in your answer sheet. Pay close attention to units—many questions test unit conversion skills. Remember that surface area is measured in square units (cm², m²) while volume is measured in cubic units (cm³, m³, litres). When dealing with cylinders and cones, always confirm whether the question provides radius or diameter. For spheres and hemispheres, note whether you are asked for curved surface area or total surface area, as the formulae differ. These MCQs mirror the style of questions appearing in the first section of CBSE Class 9 term exams and help build speed and accuracy under time pressure. Practice these without looking at the answers first.

Q1. The total surface area of a cube with edge 5 cm is: (a) 100 cm² (b) 125 cm² (c) 150 cm² (d) 175 cm²
Q2. A cylinder has radius 3.5 cm and height 8 cm. Its curved surface area (use π = 22/7) is: (a) 88 cm² (b) 132 cm² (c) 176 cm² (d) 220 cm²
Q3. The volume of a cone with radius 6 cm and height 7 cm (π = 22/7) is: (a) 132 cm³ (b) 264 cm³ (c) 396 cm³ (d) 528 cm³
Q4. If the radius of a sphere is doubled, its volume becomes: (a) 2 times (b) 4 times (c) 6 times (d) 8 times
Q5. The slant height of a cone with radius 5 cm and vertical height 12 cm is: (a) 7 cm (b) 10 cm (c) 13 cm (d) 17 cm
Q6. A hemisphere has radius 7 cm. Its total surface area (π = 22/7) is: (a) 308 cm² (b) 462 cm² (c) 616 cm² (d) 154 cm²

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

Complete each statement by filling in the blank with the correct term, number, or formula. This section tests direct recall of formulae and key concepts from NCERT Class 9 Mathematics Chapter 11. Write your answers clearly, including units where necessary. Remember that surface area always uses square units and volume uses cubic units. When a formula involves π, use 22/7 unless the question specifies otherwise or asks for an answer in terms of π. These fill-in-the-blank questions are common in CBSE school exams and help reinforce memorisation of standard results. Students often lose easy marks here due to careless mistakes or incorrect units, so double-check your answers before moving on to the next section. This section should take no more than ten minutes if you have studied the chapter thoroughly and revised all formulae the night before your exam.

Q7. The lateral surface area of a cuboid with dimensions 8 cm × 6 cm × 5 cm is __________ cm².
Q8. The volume of a cylinder is given by the formula __________.
Q9. If the radius of a sphere is 'r', its surface area is __________.
Q10. The curved surface area of a hemisphere with radius 14 cm (π = 22/7) is __________ cm².
Q11. A cone and a cylinder have the same base radius and height. The ratio of their volumes is __________.

Section C: Match the Following / True or False (1 mark each)

Match each item in Column A with the correct corresponding item in Column B. Write the pair (e.g., 1–d, 2–a) in your answer sheet. This section tests your ability to connect shapes with their correct formulae and properties. Pay attention to subtle differences: for example, curved surface area versus total surface area, or the factor of ⅓ in cone volume versus full πr²h in cylinder volume. These matching questions often appear in CBSE school unit tests and serve as quick concept checks. Students should be able to complete this section in under eight minutes. If you are unsure, use the process of elimination—cross out pairs you are certain about, then logically deduce the remaining matches. Alternatively, this section may contain five True/False statements; mark T or F and justify briefly if required by your teacher. Always read the statement carefully before marking, as small words like 'always', 'never', or 'only' can change the truth value entirely.

Column A: (1) Surface area of cube; (2) Volume of sphere; (3) CSA of cylinder; (4) Volume of cone; (5) TSA of hemisphere
Column B: (a) 2πrh; (b) 6a²; (c) ⅓πr²h; (d) ⅘πr³; (e) 3πr²
Alternatively: T/F: The volume of a cone is one-third that of a cylinder with the same base and height. (True)
T/F: Doubling the radius of a sphere doubles its surface area. (False—it becomes 4 times)
T/F: A cuboid and a cube can have the same volume but different surface areas. (True)

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

Answer the following questions in two to three steps, showing all working clearly. Each question carries 2 or 3 marks as indicated. Marks are awarded for correct method even if the final answer has a minor arithmetic error, so always write formulae first, substitute values, then compute. Use π = 22/7 unless stated otherwise. Remember to include units in your final answer; omitting units can cost you half a mark. These questions are typical of what appears in the CBSE Class 9 term-1 and term-2 exams and require solid understanding of how to apply the standard formulae. Time management is key—spend no more than 4–5 minutes per question. If you get stuck, move on and return later. This section builds confidence and ensures you can handle the bulk of the board exam paper, which consists of similar short-answer items worth two to three marks each. Practice these daily for best results.

Q12. (2 marks) Find the total surface area of a cuboid with length 10 cm, breadth 8 cm, and height 6 cm.
Q13. (3 marks) A cylindrical pillar has diameter 56 cm and height 3.5 m. Find the cost of painting its curved surface at ₹15 per m². (Use π = 22/7)
Q14. (2 marks) The radius of a sphere is 10.5 cm. Find its volume. (π = 22/7)
Q15. (3 marks) A conical tent has base radius 7 m and slant height 25 m. Find the canvas required to make the tent.
Q16. (3 marks) A hemispherical bowl is made of steel 0.5 cm thick. Inner radius is 6 cm. Find the volume of steel used.

Section E: Long Answer / HOTS Questions (4–5 marks each)

These three questions test higher-order thinking skills (HOTS) and multi-step problem solving. Each carries 4 or 5 marks and may involve combining two or more shapes, converting units, or applying real-world contexts. Write your solution step-by-step, clearly stating assumptions and showing all intermediate calculations. Marks are distributed for method, correct substitution, and final answer. CBSE examiners look for logical flow—if you arrive at the wrong answer but your method is sound, you can still earn 60–70 per cent of the marks. These questions often appear in Section C or D of the CBSE board paper and separate average students from high scorers. Spend 8–10 minutes per question and use diagrams where helpful. If the problem involves a composite solid (like a cylinder with a cone on top), find the surface area or volume of each part separately, then combine as required. Practice similar problems from NCERT exemplar and previous years' board papers to build speed and confidence.

Q17. (5 marks) A toy is in the form of a cone mounted on a hemisphere. The radius of the base is 3.5 cm and the total height is 15.5 cm. Find the total surface area of the toy. (π = 22/7)
Q18. (4 marks) A cylindrical water tank has internal diameter 10 m and height 6 m. If 1 m³ holds 1000 litres, how many litres of water can it hold when full? Also find the cost of painting its inner curved surface at ₹50 per m².
Q19. (5 marks) A solid metallic sphere of radius 10.5 cm is melted and recast into smaller cones, each of radius 3.5 cm and height 3 cm. Find the number of cones formed. (π = 22/7)

Case-Study Question (4 marks)

Read the case study carefully and answer the sub-questions that follow. This format was introduced by CBSE in 2020-21 and now appears regularly in Class 9 term exams. The case-study integrates a real-world scenario—such as construction, packaging, water storage, or manufacturing—with mathematical modelling using surface areas and volumes. Marks are typically distributed as 1+1+2 or 1+1+1+1 across three or four sub-parts. Read the passage twice, underline key data (dimensions, rates, costs), and identify which solid shapes are involved. Draw a rough diagram if it helps visualise the problem. Show all working for calculation-based sub-parts and write answers in the context of the question, not just as naked numbers. This question type rewards careful reading and the ability to apply Chapter 11 concepts to practical situations, mirroring how engineers, architects, and designers use mensuration daily in their work. Parents in cities like Bengaluru, Pune, and Hyderabad report that students who practice case-studies score 3–4 marks higher overall.

Case: A municipal corporation is building a cylindrical water tank with a hemispherical top. The cylindrical portion has radius 3.5 m and height 8 m. The hemispherical dome has the same radius.
(i) (1 mark) Find the total height of the tank from ground to the top of the dome.
(ii) (1 mark) Calculate the curved surface area of the cylindrical portion. (π = 22/7)
(iii) (2 marks) Find the total volume of water the tank can hold when completely filled. (π = 22/7)

Complete Answer Key with Explanations

This section provides full solutions to all questions. Students should attempt the worksheet independently first, then check answers here. For MCQs and fill-in-the-blanks, brief working is shown. For short and long answers, step-by-step solutions are given with key formulae highlighted. If your answer differs, recheck unit conversions and ensure you used the correct formula. Common mistakes include using diameter instead of radius, forgetting to convert cm to m, and confusing curved surface area with total surface area. When calculating volumes, always cube the radius correctly—students often write r² instead of r³ for spheres. For composite solids, ensure you add or subtract volumes/areas as required by the problem. Use this answer key not just to tick right or wrong, but to understand the logic and method behind each solution. Discuss tricky questions with classmates or teachers, and note down any formula you had forgotten for quick revision before exams. CBSETUTOR.ai offers a 24×7 AI tutor where students can upload photos of worksheet problems they find difficult and receive instant step-by-step solutions, all for a flat ₹999/month across classes 6–12, with a 3-day free trial to explore the platform.

Section A Answers: Q1(c), Q2(c), Q3(b), Q4(d), Q5(c), Q6(b)
Section B Answers: Q7: 148, Q8: πr²h, Q9: 4πr², Q10: 616, Q11: 1:3
Section C Answers: 1–b, 2–d, 3–a, 4–c, 5–e
Section D detailed solutions follow below
Section E detailed solutions follow below
Case-study detailed solutions follow below

Section D Detailed Solutions

Q12: TSA of cuboid = 2(lb + bh + hl) = 2(10×8 + 8×6 + 6×10) = 2(80 + 48 + 60) = 376 cm². Q13: Diameter = 56 cm ⇒ radius = 28 cm = 0.28 m; height = 3.5 m. CSA = 2πrh = 2×(22/7)×0.28×3.5 = 2×22×0.04×3.5 = 6.16 m². Cost = 6.16 × 15 = ₹92.40. Q14: Volume of sphere = ⅘πr³ = (4/3)×(22/7)×(10.5)³ = (4/3)×(22/7)×1157.625 = 4851 cm³. Q15: Canvas = CSA of cone = πrl = (22/7)×7×25 = 22×25 = 550 m². Q16: Outer radius = 6 + 0.5 = 6.5 cm. Volume of steel = ⅔π(R³ − r³) = (2/3)×(22/7)×(6.5³ − 6³) = (2/3)×(22/7)×(274.625 − 216) = (2/3)×(22/7)×58.625 ≈ 122.5 cm³. These solutions show all intermediate steps; ensure you write them in your exam answer sheet to earn full method marks even if arithmetic slips occur. If a question asks for cost or real-world quantity, always write the final answer with correct units and context.

Section E and Case-Study Detailed Solutions

Q17: Hemisphere radius = cone base radius = 3.5 cm. Cone height = 15.5 − 3.5 = 12 cm. Slant height l = √(3.5² + 12²) = √(12.25 + 144) = √156.25 = 12.5 cm. TSA = CSA of cone + CSA of hemisphere = πrl + 2πr² = πr(l + 2r) = (22/7)×3.5×(12.5 + 7) = 11×19.5 = 214.5 cm². Q18: Radius = 5 m. Volume = πr²h = (22/7)×25×6 = (22/7)×150 = 471.43 m³ ≈ 471430 litres. CSA = 2πrh = 2×(22/7)×5×6 = (44/7)×30 = 188.57 m². Cost = 188.57×50 ≈ ₹9428.57. Q19: Volume of sphere = (4/3)×(22/7)×(10.5)³ = 4851 cm³. Volume of one cone = (1/3)×(22/7)×(3.5)²×3 = (1/3)×(22/7)×12.25×3 = 38.5 cm³. Number of cones = 4851 / 38.5 = 126. Case-study: (i) 11.5 m. (ii) 176 m². (iii) Volume = (22/7)×(3.5)²×8 + (2/3)×(22/7)×(3.5)³ = 308 + 89.83 ≈ 397.83 m³. These detailed answers help students understand multi-step reasoning and the importance of writing every formula and substitution clearly for maximum marks in board exams.

Frequently asked questions

What is the best way to memorise all the formulae for Surface Areas and Volumes in Class 9?+

Create a one-page formula sheet with diagrams of each solid (cube, cuboid, cylinder, cone, sphere, hemisphere) and write TSA, CSA, and volume formulae beside each. Revise this sheet daily for 10 minutes. Practice at least 5 problems per shape to reinforce memory through application.

How many marks does Chapter 11 carry in the CBSE Class 9 final exam?+

Surface Areas and Volumes typically carries 9–12 marks in the CBSE Class 9 Mathematics annual exam, distributed across 2–3 short-answer (2–3 marks each) and 1–2 long-answer (4–5 marks) questions. One case-study sub-part may also appear.

Why do I keep confusing radius and diameter in cylinder and cone problems?+

Always underline or highlight whether the question gives radius (r) or diameter (d). Write 'r = d/2' at the top of your working if diameter is given. This simple habit prevents the most common error in mensuration problems and saves at least 2–3 marks per exam.

Should I use π = 22/7 or π = 3.14 in the worksheet and exams?+

Use π = 22/7 unless the question explicitly states otherwise or asks for decimal approximation. CBSE marking schemes often expect 22/7 for exact fractional answers. If the problem says 'take π = 3.14', follow that instruction to match the answer key.

How can I improve speed in solving surface area and volume questions?+

Practise mental multiplication of common numbers like 22×7, 22×14, 22×21. Memorise cubes up to 15³ and squares up to 30². Use shortcuts: for example, if r = 7 cm and π = 22/7, then πr = 22 directly, simplifying many calculations instantly.

What is the difference between curved surface area and total surface area?+

Curved surface area (CSA) includes only the curved or lateral surfaces, excluding top and bottom faces. Total surface area (TSA) includes all surfaces. For a cylinder, CSA = 2πrh; TSA = 2πrh + 2πr². Always read the question carefully to identify which is required.

Are composite solid problems likely to appear in the Class 9 board exam?+

Yes, composite solids (e.g., a cone on a cylinder or hemisphere) regularly appear as 4–5 mark long-answer questions. Break the problem into parts: find surface area or volume of each solid separately, then add or subtract as needed. Draw a labelled diagram to avoid missing any surface.

How does CBSETUTOR.ai help with difficult Surface Areas and Volumes problems?+

CBSETUTOR.ai provides a 24×7 AI tutor that accepts photo uploads of any problem. Students get instant step-by-step solutions with explanations in simple language. At ₹999/month flat for classes 6–12, with a 3-day free trial, it is an affordable way to clear doubts anytime without waiting for tuition class.

What are common mistakes to avoid in this chapter?+

Common errors include using diameter instead of radius, forgetting to convert units (cm to m or vice versa), confusing slant height with vertical height in cones, writing r² instead of r³ in volume formulae, and omitting units in the final answer. Double-check every substitution before calculating.

Is it necessary to learn derivations of formulae for CBSE Class 9 exams?+

CBSE Class 9 exams do not ask for derivations of surface area and volume formulae; those appear in Class 10. However, understanding why πr²h is cylinder volume (area of base × height) aids retention and prevents confusion. Focus on application, not proof, for Class 9.

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