Chapter Hub
Surface Areas and Volumes
This chapter helps you work with the surface area and volume of common solids such as cylinders, cones, spheres, and hemispheres. The key exam habit is to identify the correct part of the solid first: curved surface, total surface, or full volume. Many mistakes happen when you mix radius, height, and slant height, so every formula must be used with care and proper units. The chapter also includes recasting and combination of solids, which are very important in CBSE Class 10 questions. In such problems, volume is usually conserved when material is reshaped, while surface area problems require attention to exposed parts only. A strong grip on formulas, diagrams, and common mistakes can turn this chapter into a scoring one.
Difficulty
Medium
Study time
96-120 min
Plan by time
Pick the window that matches what you have right now.
If you have 15 min
Last-pass revision
Skim the Quick Revision table — definitions, formulas, and the traps board examiners reuse.
Open Quick RevisionIf you have 45 min
Targeted practice
Read the high-priority concepts, then take the chapter MCQ quiz to find weak spots.
Start MCQ QuizIf you have 96 min
First full pass
Walk every concept in chapter order, then revise and quiz. Best for the first time you study this chapter.
Open Key ConceptsChapter Learning Map
Start with one of the buckets below, then open the full map when you want the complete concept roadmap.
Cylinder
Covers Curved surface area of a cylinder, Total surface area of a cylinder and Volume of a cylinder.
Open mapCone
Covers Curved surface area of a cone, Total surface area of a cone and Volume of a cone.
Open mapSphere and Hemisphere
Covers Surface area of a sphere, Volume of a sphere, Surface area of a hemisphere and Volume of a hemisphere.
Open mapMixed Solid Problems
Covers Conversion of one solid into another and Combination of solids.
Open mapKey Concepts
Concepts grouped the way the chapter is taught — open the group that matches what you want to revise.
Cylinder
Covers Curved surface area of a cylinder, Total surface area of a cylinder and Volume of a cylinder.
Curved surface area of a cylinder
The curved surface area of a cylinder is the area of only the side surface, excluding both circular bases.
Total surface area of a cylinder
The total surface area of a cylinder is the area of its curved surface plus the areas of both circular bases.
Volume of a cylinder
The volume of a cylinder is the amount of space enclosed inside it.
Cone
Covers Curved surface area of a cone, Total surface area of a cone and Volume of a cone.
Curved surface area of a cone
The curved surface area of a cone is the area of its sloping side, excluding the circular base.
Total surface area of a cone
The total surface area of a cone is the area of its curved surface plus the area of its circular base.
Volume of a cone
The volume of a cone is one-third of the volume of a cylinder having the same base radius and height.
Sphere and Hemisphere
Covers Surface area of a sphere, Volume of a sphere, Surface area of a hemisphere and Volume of a hemisphere.
Surface area of a sphere
The surface area of a sphere is the total area of its outer curved surface.
Volume of a sphere
The volume of a sphere is the amount of space enclosed inside it.
Surface area of a hemisphere
The surface area of a hemisphere is the curved surface plus the circular base when the whole outer surface is asked for.
Volume of a hemisphere
The volume of a hemisphere is half the volume of a sphere with the same radius.
Mixed Solid Problems
Covers Conversion of one solid into another and Combination of solids.
Conversion of one solid into another
When one solid is melted, recast, or reshaped into another solid without wastage, the volume of material remains the same.
Combination of solids
A combination of solids is a shape formed by joining two or more simple solids together.
Exam Intelligence
The traps that cost marks in this chapter, and how its main concepts turn up in the paper.
High Probability Topics
- Curved surface area of a cylinder
- Total surface area of a cylinder
- Volume of a cylinder
- Curved surface area of a cone
- Total surface area of a cone
- Volume of a cone
- Surface area of a sphere
- Volume of a sphere
Common Traps
- Using total surface area when only curved surface area is asked.
- Forgetting to square or cube the radius in volume formulas.
- Using slant height instead of perpendicular height in cone volume.
- Counting hidden joined faces in composite solids.
- Mixing surface area with volume in recasting questions.
- Using the wrong hemisphere formula when the base is included.
How CBSE Asks This Chapter
- Curved surface area of a cylinder
Find the curved surface area of a cylindrical container, pipe, drum, or label region using the given radius and height.
- Total surface area of a cylinder
Find the total outer area of a closed cylindrical can, drum, tank, or pipe end-to-end surface.
- Volume of a cylinder
Find the capacity of a cylindrical tank, drum, pipe section, or solid cylinder from given dimensions.
- Curved surface area of a cone
Find the area of the conical side of a tent, cap, funnel, or ice-cream cone from radius and slant height.
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
Curved surface area of a cylinder
Remember
Curved surface area of a cylinder = 2πrh, where r is the radius of the base and h is the height of the cylinder.Trap to avoid
Students often use the total surface area formula and include the two circular ends even when only the side is required.
Total surface area of a cylinder
Remember
Total surface area of a cylinder = 2πr(h+r), which is the curved surface area 2πrh plus the two base areas 2πr².Trap to avoid
Students often forget one or both circular bases and write only the curved surface area.
Remember
Volume of a cylinder = πr²h, where r is the radius of the base and h is the height.Trap to avoid
Students often use surface area formulas instead of the volume formula, or they forget to square the radius.
Remember
Curved surface area of a cone = πrl, where r is the radius of the base and l is the slant height.Trap to avoid
Students often use the vertical height instead of the slant height in the formula.
Remember
Total surface area of a cone = πr(l+r), where r is the radius and l is the slant height.Trap to avoid
Students often calculate only the curved surface area and forget to add the circular base.
Remember
Volume of a cone = 1/3 πr²h, where r is the radius of the base and h is the perpendicular height.Trap to avoid
Students often forget the one-third factor or use slant height instead of perpendicular height.
Remember
Surface area of a sphere = 4πr², where r is the radius of the sphere.Trap to avoid
Students often use 2πr² or treat the sphere like a hemisphere or cylinder.
Remember
Volume of a sphere = 4/3 πr³, where r is the radius of the sphere.Trap to avoid
Students often use 4πr², which is the surface area formula, instead of the volume formula.
Remember
Curved surface area of a hemisphere = 2πr².Trap to avoid
Students often use only 2πr² and forget that the circular base must also be included for total surface area.
Remember
Volume of a hemisphere = 2/3 πr³, where r is the radius of the hemisphere.Trap to avoid
Students often write one-half of 4/3 πr³ as 1/3 πr³ instead of 2/3 πr³.
Conversion of one solid into another
Remember
For conversion problems without wastage, volume of the original solid = volume of the new solid.Trap to avoid
Students often compare surface areas instead of volumes, or they forget that the total material stays the same.
Remember
For a combination of solids, total volume = sum of component volumes.Trap to avoid
Students often count hidden joining surfaces or forget to add one of the component volumes.
| Concept | Remember | Trap to avoid |
|---|---|---|
| Curved surface area of a cylinder | Curved surface area of a cylinder = 2πrh, where r is the radius of the base and h is the height of the cylinder. | Students often use the total surface area formula and include the two circular ends even when only the side is required. |
| Total surface area of a cylinder | Total surface area of a cylinder = 2πr(h+r), which is the curved surface area 2πrh plus the two base areas 2πr². | Students often forget one or both circular bases and write only the curved surface area. |
| Volume of a cylinder | Volume of a cylinder = πr²h, where r is the radius of the base and h is the height. | Students often use surface area formulas instead of the volume formula, or they forget to square the radius. |
| Curved surface area of a cone | Curved surface area of a cone = πrl, where r is the radius of the base and l is the slant height. | Students often use the vertical height instead of the slant height in the formula. |
| Total surface area of a cone | Total surface area of a cone = πr(l+r), where r is the radius and l is the slant height. | Students often calculate only the curved surface area and forget to add the circular base. |
| Volume of a cone | Volume of a cone = 1/3 πr²h, where r is the radius of the base and h is the perpendicular height. | Students often forget the one-third factor or use slant height instead of perpendicular height. |
| Surface area of a sphere | Surface area of a sphere = 4πr², where r is the radius of the sphere. | Students often use 2πr² or treat the sphere like a hemisphere or cylinder. |
| Volume of a sphere | Volume of a sphere = 4/3 πr³, where r is the radius of the sphere. | Students often use 4πr², which is the surface area formula, instead of the volume formula. |
| Surface area of a hemisphere | Curved surface area of a hemisphere = 2πr². | Students often use only 2πr² and forget that the circular base must also be included for total surface area. |
| Volume of a hemisphere | Volume of a hemisphere = 2/3 πr³, where r is the radius of the hemisphere. | Students often write one-half of 4/3 πr³ as 1/3 πr³ instead of 2/3 πr³. |
| Conversion of one solid into another | For conversion problems without wastage, volume of the original solid = volume of the new solid. | Students often compare surface areas instead of volumes, or they forget that the total material stays the same. |
| Combination of solids | For a combination of solids, total volume = sum of component volumes. | Students often count hidden joining surfaces or forget to add one of the component volumes. |
Practice
Use short concept checks first, then move into the full chapter test.
Free Chapter MCQ Quiz
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