Class 10 Surface Area and Volume of Combined Solids
Painting, filling and melting describe different quantities. Identify the operation first: paint covers a surface, capacity measures space, and recasting usually preserves volume.
Sources checked 2 October 2026 · original study guide · CBSE/NCERT context
Method guide · Read at your own pace
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| Task | Quantity | First check |
|---|---|---|
| Paint a joined solid | Exposed surface area | Which faces are hidden at the join? |
| Fill a container | Internal volume | Are dimensions internal or external? |
| Melt and recast a solid | Equal total volumes if no loss | Are all dimensions in the same unit? |
Mark the hidden face before adding areas
Trace the outside outline and list the two curved surfaces plus the bottom disk. The dashed circle marks the join; it is not an extra surface to paint. Radius and cylinder height belong to different formula terms.
A cylinder joined to a hemisphere
Imagine a cylinder of radius 3 cm and height 8 cm, capped by a hemisphere of the same radius. Its bottom is exposed. The circular faces at the join are internal and should not be counted in the outside area.
The curved cylinder contributes 2πrh = 48π cm²; the hemisphere contributes 2πr² = 18π cm²; the bottom contributes πr² = 9π cm². Total exposed area is 75π cm². Adding the full surface areas of both solids would count the joined circle twice.
Use a different sum for volume
Volume is additive here because the solids meet at a face and their interiors do not overlap. Cylinder volume is πr²h = 72π cm³ and hemisphere volume is ⅔πr³ = 18π cm³. Total volume is 90π cm³. There is no face area to subtract from a volume.
For a hollow object, subtraction may be needed: outer volume minus inner volume gives material volume. Keep “space inside” distinct from “material used”. Sketch both boundaries before putting dimensions into a formula.
A practical checking routine
Circle the required unit in your answer: cm² for area or cm³ for volume. List each exposed surface once, then compare that list with the terms in your expression. Check whether a base is open or closed. A container described as open at the top does not have a top disk.
If converting units, convert lengths before calculating. Alternatively, remember that 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³. Multiplying an area or volume by 100 merely because metres became centimetres is a dimensional error.
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Sources and scope
This is a focused learning resource, not a complete chapter or a prediction of board questions. Use your current syllabus and the paper-specific marking scheme for assessment requirements.
Frequently asked questions
Do I always subtract the joining area twice?
Only if you first added total surface areas that each included that face. Listing the exposed surfaces directly is usually clearer.
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