Chapter 14.2 — Exercise 14.1 — Surface Area

A Close Call, Decided Only by Calculating

14,800 is smaller than 15,000, so the cuboid needs slightly less material than the cube — a useful reminder that "which shape uses less material" isn't something that can be guessed just by comparing overall size; it has to be worked out from the actual surface area formula, since these two boxes are close enough in size that intuition alone wouldn't reliably say which one wins.

It's worth noticing just how close these two totals actually are — 14,800 against 15,000, a gap of only 200 sq. units out of roughly 15,000, less than a 1.5% difference. That closeness isn't an accident: both boxes were deliberately chosen with very similar overall dimensions (the cuboid's largest and smallest sides, 60 and 40, average out close to the cube's uniform 50), which is exactly why guessing the answer without calculating would have been unreliable here. If the two boxes had been very different in shape — a long thin cuboid against a compact cube of similar volume, say — the surface areas would likely have differed by far more than 1.5%, since a long, thin shape generally has more surface area relative to its size than a compact one does.

Working Backwards From a Known Surface Area

A cube has total surface area 600 sq. cm — what's the length of its side? Since T.S.A. = 6l², this is a straightforward reverse calculation:

Whenever a problem hands over the surface area and asks for a dimension instead of the other way around, the fix is always the same: write the formula down first, substitute what's known, and solve for whatever's missing — here, taking a square root at the very last step rather than trying to guess the side length directly. It's worth pausing on why the square root only needed to consider the positive answer: a negative side length has no physical meaning for an actual cube, so even though 100 technically has two square roots, ±10, only +10 makes sense as an answer here.

Painting Everything Except Top and Bottom

A cabinet measuring 1 m × 2 m × 1.5 m gets painted on every face except its top and bottom. That's precisely what lateral surface area was built to describe — the four "standing" faces, with the two horizontal ones deliberately excluded:

The phrase "except the top and bottom" is the signal to reach for L.S.A. rather than T.S.A. — recognising that phrase is really the entire content of this problem; the arithmetic that follows is routine once the right formula has been chosen. It's worth checking what would have gone wrong using T.S.A. by mistake: 2(lh+bh+lb) = 2(1.5+3+2) = 2(6.5) = 13 sq. m, a full 4 sq. m larger than the correct 9 sq. m — the difference being exactly twice the area of one 1 m × 2 m top-or-bottom face (2×1×2=4), which is precisely the area a T.S.A. calculation would have wrongly counted in.

Turning an Area Into a Cost

A cuboid measuring 20 cm × 15 cm × 12 cm gets painted all over (every face this time, not just the sides), at a rate of 5 paise per square centimetre. The area comes first, exactly as before, and the cost is a separate multiplication tacked on at the end:

Converting 7200 paise into rupees at the very end (dividing by 100, since 100 paise = ₹1) is a small but necessary last step — the area calculation alone only answers "how many square centimetres," not "how many rupees," and mixing up paise and rupees partway through would make the final answer a hundred times too large or too small. It's the same kind of unit discipline already practised in earlier chapters — converting centimetres to metres before applying a per-square-metre flooring rate, or minutes to hours before comparing speeds — showing up here in a currency instead of a length or a time.

Reading "Except" and "All Faces" as Instructions, Not Decoration

All four problems in this exercise use the exact same two formulas from the chapter introduction — the entire skill being tested is choosing correctly between T.S.A. and L.S.A. based on the wording. "Painted all surfaces except the top and bottom" means L.S.A.; "painting a cuboid" with no such exception, or comparing "material to make" a whole box, means T.S.A. Misreading either phrase doesn't produce an obviously wrong-looking answer — it produces a perfectly reasonable-looking number that simply answers a different question than the one actually being asked, exactly as the 13 sq. m versus 9 sq. m mix-up above demonstrates: both numbers look like sensible surface areas, and only one of them actually answers what the cabinet problem asked.

Two Formulas, Four Problems, No New Arithmetic

Looking back across all four problems in this exercise, the actual multiplication and addition involved never gets more complicated than what a calculator-free student could manage directly — the challenge in every case sits entirely in deciding which formula applies and in what order to substitute the given numbers, not in the arithmetic itself. That's a deliberate feature of this stage in the surface-area chapter: once T.S.A. and L.S.A. have been derived once, properly, from first principles in the chapter introduction, applying them correctly becomes the whole task, freeing up attention for exactly the kind of careful reading — "except the top and bottom," "at the rate of," "material required" — that these four problems were designed to test.

From Covering a Solid to Filling One

Every problem in this lesson asked how much material covers the outside of a cube or cuboid. Exercise 14.2 asks a related but distinct question about the very same shapes — not how much surface they have, but how much space they enclose — building on the volume formulas V = lbh and V = l³ that follow the same "peel apart the solid and measure its parts" logic as the surface-area formulas here, but answer a question about capacity rather than covering.