Volume and Surface Area

Volume, lateral and total surface, diagonal.

• The diagonal joins opposite corners of the whole solid, not of one face — it is the longest rod that fits in a room.
• If the areas of three adjacent faces are given, their product is $(lbh{)}^2$, so the volume is its square root.
• If a ratio is given, take edges $px,qx,rx$ and find $x$ from the given quantity.

The area of four walls, the material in a hollow box, an open box and the capacity of a tank.

• Room: four walls $=2(l+b)h$; floor or ceiling $=lb$; volume $=lbh$. Painting does not include floor and ceiling.
• Hollow box (thickness $t$): inner sizes are $2t$ less each way — $(l-2t)(b-2t)(h-2t)$; material $=$ outer $-$ inner volume.
• Open box (squares cut from the corners): volume $=(L-2x)(B-2x)x$.
• Tank capacity: volume (cu m) $\times 1000$ = litres. Time to fill $=\frac{\text{capacity}}{\text{flow rate}}$.

Cutting small cubes from a big solid, melting several cubes into one, and counting painted cubes.

• Cutting: $\frac{\text{big size}}{\text{small size}}$ cubes in each direction — multiply the three counts.
• Melting: the volume does not change. Three cubes into one: $A^3=a^3+b^3+c^3$.
• Painted cube ($n$ small cubes per edge): three faces painted — $8$ (corners); two faces — $12(n-2)$; one face — $6(n-2{)}^2$; none — $(n-2{)}^3$.

Check: the four counts must add up to $8+12(n-2)+6(n-2{)}^2+(n-2{)}^3=n^3$ — try $n=4$: $8+24+24+8=64=4^3$ ✓.

Solid and hollow cylinders, a roller, wire, and water from a pipe.

Slant height, volume, surface and the frustum.

The sphere’s volume and surface, the hemisphere, melting and marbles.

Turn each size into a multiplier, apply the right power, multiply.

Volumes add on joining, but the joined faces are no longer on the outside.

The slips that cost marks.

Habits for speed.

One-formula questions.

Exam-level questions.

Questions at the hard end of competitive exams.