Volume and Surface Area — Extra Practice

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.

• Cut a cylinder open and flatten it and you get a rectangle — length $2\pi r$, breadth $h$; that is why the curved surface is $2\pi rh$.
• Roller / wheel: one turn covers the area of its curved surface; total area $=$ curved surface $\times$ turns.
• Recasting: the volume is equal — rod to wire: $\pi r_1^2{h}_1=\pi r_2^2{h}_2$.
• Water from a pipe: volume per minute $=\pi r^2\times$ speed (all in metres).

Slant height, volume, surface and the frustum.

• A cone is one-third of the cylinder with the same base and height. That is where the $\frac{1}{3}$ comes from.
• Curved surface uses $l$ (slant), not $h$; volume uses $h$, not $l$.
• Cut a cone parallel to its base and the top small cone is similar to the whole — volumes are in the cube of the ratio of heights.

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

• A sphere’s surface is exactly four times the area of its biggest circle.
• A hemisphere’s total surface is $3\pi r^2$ — curved $2\pi r^2$ plus the flat circle $\pi r^2$.
• Melting: volume of $n$ small spheres = volume of the big one ⟹ $n={\left(\frac{R}{r}\right)}^3$.
• The volume of a submerged object = the rise in water (marbles, balls, stones).

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

• Increase $x\%$ → multiplier $1+\frac{x}{100}$; decrease $x\%$ → multiplier $1-\frac{x}{100}$.
• If every size becomes $k$ times: lengths $k$ times, surface area $k^2$ times, volume $k^3$ times.
• Cylinder: $V\propto r^{2}h$ — square the radius multiplier, take the height multiplier as it is.
• Cuboid: the volume multiplier = the product of the three multipliers.
• Keeping the volume fixed: in a cylinder, if $r$ grows then $h$ falls in proportion to its square.