Chapter at a glance
- Objects in daily life are often combinations of basic solids (cuboid, cylinder, cone, hemisphere, sphere).
- Surface area of a combined solid equals the sum of the curved surface areas of only the exposed parts; hidden or joined faces are excluded.
- Volume of a combined solid equals the sum of the volumes of the individual solids.
- When painting or covering a toy/model, only the visible outer surfaces are considered; bases resting on other solids are omitted.
- For vessels or containers, inner surface area or actual capacity accounts for depressions, raised portions or hollow parts.
- Height of one solid in a combination is found by subtracting the height/radius of the attached solid from the total height.
- When bases of cone and cylinder differ in size, the exposed ring portion of the larger base must be included in surface area.
- Capacity problems may require subtracting the volume of a hemisphere or other solid from the apparent volume.
Definitions, theorems and results
No formal theorems, lemmas or axioms requiring proof are stated in the chapter. The chapter applies previously known results on surface areas and volumes of basic solids to combinations. The key results framed by NCERT are:
- Total surface area of a new solid formed by joining two or more basic solids = sum of the curved surface areas of the exposed parts only.
- Volume of the solid formed by joining two or more basic solids = sum of the volumes of the individual solids.
- When a hemisphere is attached to a cube or cuboid, the base area of the hemisphere is subtracted from the TSA of the cube/cuboid and its CSA is added.
- When bases of different radii meet (e.g., cone on cylinder), the area to be painted includes CSA of cone + (area of base of cone − area of base of cylinder).
Formula sheet
| Formula |
Meaning of symbols |
| CSA hemisphere = 2πr² |
r = radius |
| CSA cylinder = 2πrh |
r = radius, h = height |
| CSA cone = πrl |
r = base radius, l = slant height |
| Volume cylinder = πr²h |
r = radius, h = height |
| Volume cone = (1/3)πr²h |
r = base radius, h = height |
| Volume hemisphere = (2/3)πr³ |
r = radius |
| TSA cube = 6a² |
a = edge length |
| Slant height of cone l = √(r² + h²) |
r = base radius, h = vertical height |
| Volume of air/shed = volume of cuboid + ½ volume of cylinder |
— |
Solved-example patterns
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TSA of toy/model (cone/hemisphere or cylinder/hemisphere)
Steps: Identify exposed surfaces → write TSA = CSA of hemisphere + CSA of cone/cylinder → calculate height of cone by subtracting hemispherical radius from total height → find slant height if needed → substitute values, ensuring only curved surfaces are added.
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Surface area when one solid is mounted on another with partial overlap (cube + hemisphere, cylinder + cone)
Steps: Start with TSA of larger solid → subtract area of the face covered by the smaller solid → add CSA of the smaller solid → adjust for any exposed ring if bases differ in size.
-
Painted/coloured area on different parts (rocket, top)
Steps: Separate surfaces to be painted different colours → for cone portion include CSA + (base of cone − base of cylinder) if ring is visible → for cylinder include CSA + one base if exposed.
-
Volume of combined solid or capacity of vessel
Steps: Add volumes of constituent solids (cuboid + half-cylinder, hemisphere + cone) → for capacity, subtract volume of internal solid (hemisphere) from apparent cylindrical volume.
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Volume of remaining solid or water displaced
Steps: Calculate volume of outer solid → subtract volume of removed or immersed solid(s) → apply any percentage or number-of-objects condition given.
Common mistakes and exam pitfalls
- Adding full TSA of cone and hemisphere instead of only CSA of each (internal faces are hidden).
- Forgetting to subtract the base area of the hemisphere from the cube/cuboid face.
- Using total height directly as cone height without subtracting hemispherical radius.
- Ignoring the ring area when cone base > cylinder base.
- Mixing units (cm vs m) or forgetting to convert final answer (e.g., cm² to m²).
- Taking π inconsistently within the same question when value is specified.
- Calculating volume of air without subtracting space occupied by machinery/workers.