Chapter 12 · CBSE Class 10 Maths
Surface Areas and Volumes
Two ways to learn — a crisp formula reference, or a story that makes combined solids feel obvious.
Four scenes that build the ideas from scratch, step by step.
How this works
Four scenes, one thread: when two solids join into one, surface area and volume behave in opposite ways — surface hides what's joined, but volume never loses anything. Read the story, then tap "The Maths" for the formal result.
Building the Toy
Rasheed's birthday top (lattu) is a cone stacked on a hemisphere, joined flat-face to flat-face, same radius. To colour the whole outside of the toy, which surfaces actually need paint?
The cone has a curved side AND a flat circular base. The hemisphere has a curved side AND a flat circular face too. Do all four surfaces get painted?
No — where the cone's flat base meets the hemisphere's flat face, that shared surface is now buried INSIDE the toy. Only the two curved surfaces are visible from outside.
TSA of the toy = CSA of cone + CSA of hemisphere
= πrl + 2πr² (NOT the sum of each piece's full TSA)
The Ring That Needs Painting
A wooden toy rocket is a cone mounted on a cylinder — but this time, the cone's base (2.5 cm radius) is WIDER than the cylinder's base (1.5 cm radius). Does the join hide everything cleanly, like the top in Scene 1?
Picture looking straight down at where the cone sits on the cylinder. Is there any part of the cone's base still visible around the edge?
Yes — a ring-shaped strip of the cone's base, the part sticking out beyond the narrower cylinder underneath, stays fully exposed and needs painting too.
Area to paint orange (cone side) = CSA of cone + (base area of cone − base area of cylinder)
= πrl + π(R² − r²) where R = cone's base radius, r = cylinder's base radius
Volume Doesn't Disappear
Now switch from painting (surface area) to capacity (volume). Shanta's factory shed is a cuboid topped with a half-cylinder roof. Does the same "subtract the hidden join" thinking apply to the volume of air inside?
Unlike surface area, no material or space vanishes just because two solids are stuck together. The volume is simply whatever space BOTH pieces occupy, combined.
Volume of shed = volume of cuboid + volume of half-cylinder roof
(no exclusion for the 'join' — every bit of space still counts)
This is the single most important contrast in the whole chapter: surface area calculations require carefully excluding hidden faces, but volume calculations never do — volumes of combined solids are always just added together, full stop.
The Almost-Full Glass
A juice seller's glass looks like a simple cylinder (radius 2.5 cm, height 10 cm) — but its base has a raised hemispherical bump built in, to make the glass look fuller with less actual juice.
If I just measure the outside cylinder shape, I'd expect it to hold 196.25 cm³. Does it actually hold that much?
No — the hemisphere bump physically occupies space where juice would otherwise sit, so it must be subtracted from the "apparent" cylinder volume to get the true, usable capacity.
Apparent capacity = πr²h = 3.14 × 2.5² × 10 = 196.25 cm³
Hemisphere volume = (2/3)πr³ = (2/3) × 3.14 × 2.5³ = 32.71 cm³
Actual capacity = 196.25 − 32.71 = 163.54 cm³
The Big Picture
Every idea in this chapter branches from one contrast: joining two solids hides some SURFACE (use CSA, watch for exposed rings) but never hides any VOLUME (always just add). A container that looks simple from outside can still have less usable space than it appears.
Combined surface area rule?
Sum CSAs of visible parts — never TSAs
Bases don't match at a join?
Add the exposed ring: π(R²−r²)
Combined volume rule?
Always additive — sum every piece's volume
Container with a hidden bump?
Actual capacity = apparent − bump's volume
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