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§12Class 10, Chapter 12

Surface Areas and Volumes: Combining Solids Without Losing Track of What's Hidden

Tankers, test tubes, capsules, bird-baths, and toy rockets are never just one basic solid — they're combinations. This companion covers the two opposite rules that make combination solids solvable: subtract hidden faces for surface area, simply add for volume, with tools, flashcards and a CBSE-style quiz to test yourself.

TSA = ΣCSAVolume = Σ VolumesCSA hemisphere=2πr²V cone=⅓πr²h

🧱12.1 Combining Basic Solids

From Class IX you already know four basic solids — the cuboid, cone, cylinder, and sphere — and how to find each one's surface area and volume. But real objects around you are rarely just one of these shapes in isolation.

🚛
Spot the combination.. A fuel tanker truck's container looks like a cylinder with a hemisphere capping each end. A science-lab test tube is a cylinder with one hemispherical bottom. Domes, silos, and countless everyday objects are all built from two or more basic solids stuck together.

None of these composite shapes fit neatly into any single formula you already know. This chapter's whole approach is to break a new, unfamiliar solid down into the basic pieces you've already mastered — then combine the known results in the right way.

🎨12.2 Surface Area of a Combination of Solids

Take that tanker container: a cylinder with a hemisphere stuck on each end. Once assembled, an outside observer can only ever see the curved surfaces — the two hemispheres' curved caps and the cylinder's curved side. The flat circular faces where the pieces meet are hidden inside, no longer part of the exterior at all.

hemisphere
+
cylinder
+
hemisphere
→
Capsule / Tanker
Hidden face(s): the two flat circles where each hemisphere meets the cylinder
cone
+
hemisphere
→
Toy / Lattu
Hidden face(s): the one flat circle where the cone's base meets the hemisphere's flat face
🔑The Central Rule. The total surface area (TSA) of a combination solid is the SUM OF THE CURVED SURFACE AREAS (CSAs) of its individual parts — never the sum of their total surface areas, since joining hides the flat faces where they meet.
TSA of (cylinder + 2 hemispheres) = CSA of hemisphere + CSA of cylinder + CSA of other hemisphere
🪤The single most common mistake in this chapter.. It's tempting to add the full TSA of each piece (2πr² for a hemisphere's curved part PLUS its flat circular base, and so on) — but that double-counts or wrongly includes the hidden joining faces. Always ask: which faces are still visible from OUTSIDE the finished solid?

✏️Worked Examples

A playing top is shaped like a cone surmounted by a hemisphere. Total height 5 cm, diameter 3.5 cm. Find the area to be coloured. (π = 22/7)

HEMISPHERE'S CSACSA of hemisphere = 2πr² = 2×(22/7)×1.75² ≈ 19.25 cm² (using r = 3.5/2 = 1.75 cm)
FIND THE CONE'S HEIGHT AND SLANT HEIGHTCone height = total height − radius of hemisphere = 5 − 1.75 = 3.25 cm. Slant height l = √(r²+h²) = √(1.75²+3.25²) ≈ 3.7 cm
CONE'S CSACSA of cone = πrl = (22/7)×1.75×3.7 ≈ 20.35 cm²
ADD THE TWO CSAsTotal area to colour = 19.25 + 20.35 ≈ 39.6 cm²
⚠️Notice what we did NOT do.. "Total surface area of the top" is NOT the sum of the separate total surface areas of the cone and the hemisphere — that would wrongly count the joined flat circular face twice.

📦12.3 Volume of a Combination of Solids

Surface area calculations needed care because joining solids hides some flat faces. Volume works completely differently: no material vanishes when two solids are glued together, so the total volume is simply the straightforward SUM of each part's volume.

🔑The Central Rule. The volume of a combination solid is always the SUM of the volumes of its individual parts — with no adjustment needed, unlike surface area.
↔️Surface area vs. volume: opposite instincts.. For surface area, you must SUBTRACT the hidden joining faces. For volume, you simply ADD — the space each piece occupies stays fully occupied even after joining. Mixing up these two opposite rules is a very common exam slip.

✏️Worked Examples

A shed is a cuboid (15 m × 7 m × 8 m) topped by a half-cylinder (diameter 7 m, length 15 m). Find the air volume, then subtract machinery (300 m³) and 20 workers (0.08 m³ each). (π = 22/7)

ADD THE TWO VOLUMESVolume = (cuboid) + ½(cylinder) = (15×7×8) + ½×(22/7)×3.5²×15 = 840 + 288.75 = 1128.75 m³
SUBTRACT MACHINERY AND WORKERSOccupied space = 300 + (20×0.08) = 301.6 m³. Remaining air = 1128.75 − 301.6 = 827.15 m³

📝12.4 Chapter Summary

  1. Surface area of a combination solid = the sum of the CURVED surface areas of its individual parts (never the sum of their total surface areas), since joining hides the flat faces where the pieces meet.
  2. Volume of a combination solid = the straightforward SUM of the volumes of its individual parts — no adjustment needed, since no material is lost when solids are joined.
🏗️
One toolkit, endless real shapes.. Once you can spot which basic solids make up a composite object, and remember which rule (subtract for area, add for volume) applies, you can handle tankers, test tubes, capsules, bird-baths, rockets, domes, and far more.