NCERT Solutions for Class 10 Maths Chapter 12: Surface Areas and Volumes
NCERT Solutions for Class 10 Maths Chapter 12 – Surface Areas and Volumes – is one of the highest-weightage chapters in the CBSE Class 10 board examination. This chapter takes the study of three-dimensional shapes to an advanced level, introducing students to combinations of solids and the real-life conversion of one solid into another. For other chapters of NCERT Solutions for Class 10 Maths, check NCERT Solutions for Class 10 and NCERT Solutions.
In this chapter, you will learn how to find the surface area and volume of solids like cylinders, cones, spheres, hemispheres, and frustums, as well as combinations of two or more solids. The practical angle of this chapter—problems about water tanks, tents, capsules, ice cream cones, and decorative items—makes it one of the most relatable chapters in Class 10 Maths. Myclass24 ensures that every solution is presented clearly, using diagrams wherever necessary, so that students grasp the logic and can reproduce the steps confidently during their board exams. Our PDF solutions are accessible on mobile and desktop devices for ease of revision.
Download PDF: NCERT Solutions for Class 10 Maths Chapter 12 – Surface Areas and Volumes
The free PDF of NCERT Solutions for Class 10 Maths Chapter 12 is available on Myclass24. It includes complete solutions for all four exercises—Exercise 12.1, 12.2, 12.3, and 12.4—along with solved examples from the NCERT textbook. The PDF layout is clean, mobile-friendly, and organized by exercise so you can jump to the section you need quickly. It is ideal for last-minute revision and for understanding the step-by-step method expected by CBSE examiners.
Chapter 12 – Key Concepts, Formulas, and Exercise Breakdown
Chapter 12 of NCERT Class 10 Maths is part of the Mensuration unit. It expands on the concepts studied in earlier classes and introduces new solids such as the frustum of a cone. The chapter is divided into four exercises covering surface areas of combinations, volumes of combinations, conversion of solids, and problems based on the frustum.
Complete Formula Reference Table
| Solid | Total Surface Area | Volume |
| Cube | 6a² | a³ |
| Cuboid | 2(lb + bh + hl) | l × b × h |
| Cylinder | 2πr(h + r) | πr²h |
| Cone | πr(l + r) | (1/3)πr²h |
| Sphere | 4πr² | (4/3)πr³ |
| Hemisphere | 3πr² | (2/3)πr³ |
| Frustum | π(R + r)l + π(R² + r²) | (πh/3)(R² + r² + Rr) |
Exercise-Wise Distribution of Questions
| Exercise | Topic | No. of Qs | Key Focus |
| Ex 12.1 | Surface Area of Combinations | 9 | SA of combined solids |
| Ex 12.2 | Volume of Combinations | 8 | Volumes of combined solids |
| Ex 12.3 | Conversion of Solids | 9 | Melting/recasting problems |
| Ex 12.4 | Frustum of a Cone | 5 | Frustum SA and Volume |
Important Facts and Board Exam Insights
Chapter 12 regularly contributes 5 to 7 marks in the CBSE board exam, making it one of the most valuable chapters to prepare thoroughly. Key problem types include: surmounting a cone on a cylinder or hemisphere, finding the volume of water that overflows when a solid is dropped into a vessel, converting a cylinder into a cone or sphere, and finding slant height and curved surface area of a frustum used in bucket or funnel problems. Myclass24 solutions break each problem type into a systematic method that students can follow reliably in any exam.
Conversion of Solids – A Special Focus
One of the most interesting and frequently tested topics in Chapter 12 is the conversion of solids—problems where a metallic solid is melted and recast into another shape. The key principle is that volume remains constant during melting. For example, if a metallic sphere is melted to form small cones, the total volume of all cones equals the volume of the sphere. Myclass24 explains this concept with solved examples and exercises so students can handle any variation of this problem confidently.
NCERT Solutions for Class 10 Maths Chapter 12: Surface Areas and Volumes
Surface Areas and Volumes help measure the space occupied by three-dimensional objects and the area covering their surfaces. This chapter introduces formulas for cubes, cuboids, cylinders, cones, spheres, and hemispheres. Understanding these concepts is important because they have practical applications in construction, packaging, manufacturing, and engineering. Students learn how to calculate capacities, dimensions, and material requirements for various objects. Since this chapter combines geometry with real-life applications, it is considered one of the most useful and scoring topics in Class 10 Mathematics.
Surface area refers to the total area covering the outer surface of a three-dimensional object, while volume measures the space occupied inside the object. Surface area is expressed in square units, whereas volume is expressed in cubic units. Understanding this difference is essential because examination questions often require students to determine which measurement is needed. Confusing the two concepts can lead to incorrect answers. Mastering this distinction helps students solve practical and theoretical problems involving three-dimensional figures accurately.
This chapter involves several geometric solids, each having its own formula for surface area and volume. Students must understand when and how to apply the correct formula. Memorizing formulas alone is not sufficient; understanding the meaning of dimensions such as radius, height, and length is equally important. Proper use of formulas simplifies calculations and improves problem-solving speed. Since many board examination questions are formula-based, strong command of these formulas is essential for scoring well.
Curved surface area includes only the curved portion of a solid, excluding its flat surfaces. Total surface area includes all exposed surfaces of the object, including bases and curved regions. Understanding this distinction is important because different questions require different measurements. Students often encounter problems involving cylinders, cones, and spheres where they must identify the required surface area correctly. Careful interpretation of the question helps avoid mistakes and ensures accurate calculations.
Some problems involve figures formed by combining two or more solids. In such cases, students calculate individual surface areas or volumes and then add or subtract them as required. These questions test conceptual understanding and analytical thinking. Students must carefully identify overlapping surfaces and hidden regions. Regular practice helps develop the ability to visualize complex shapes and apply formulas correctly. Combination-of-solids questions are common in examinations because they assess deeper understanding of geometric concepts.
Volume calculations are used in designing water tanks, storage containers, pipelines, packaging materials, and buildings. Engineers and architects rely on volume measurements to estimate capacities and material requirements. Understanding volume helps students appreciate how mathematics is applied in everyday life and professional fields. These applications demonstrate the practical relevance of geometry and measurement concepts. Learning such uses also makes the chapter more interesting and meaningful.
Students often use incorrect formulas, forget unit conversions, or confuse radius with diameter. Another common mistake is calculating total surface area when only curved surface area is required. Errors in arithmetic calculations can also affect final answers. To avoid these problems, students should carefully identify the given dimensions and required measurements before starting calculations. Practicing a variety of problems helps improve accuracy and confidence in solving mensuration questions.