Mensuration (3D)

The three measures

For a cylinder of radius $r$, height $h$: $V = \pi r^2 h,\qquad \text{CSA (curved)} = 2\pi r h,\qquad \text{TSA} = 2\pi r(h + r)$ The curved surface unrolls into a rectangle of width $2\pi r$ (the circumference) and height $h$ — that picture explains every formula. The two circular lids add $\pi r^2$ each for the total.

Reverse and ratio work

• From volume: $h = \frac{V}{\pi r^2}$; from CSA: $r = \frac{\text{CSA}}{2\pi h}$.
• Changing dimensions: $V \propto r^2 h$, so doubling $r$ quadruples $V$, doubling $h$ doubles it; radius $\times 2$ AND height $\div 2$ leaves $V \times 2$. These multiplier questions are one-line proportion bookkeeping.

Pipes, wells and flowing water

• Hollow pipe (tube): metal volume $= \pi (R^2 - r^2) \times$ length — the annulus times the length.
• Well/tank capacity: compute $V$ in $\text{m}^3$; litres $= 1000 V$.
• Water flow: volume delivered per second $= \pi r^2 \times$ (speed); multiply by seconds for the total. Keep the speed units (cm/s) consistent with the radius units.

Melting and recasting

Melting preserves VOLUME: a cylinder recast into a sphere/wire/spheres keeps $\pi r^2 h$ (with the same metal). Drawn-wire questions: the wire is a thin cylinder of equal volume; if the wire's radius is given as a decimal (0.1 cm etc.), be careful with the fourth power ($r^2$ on both sides).

The 22/7 number family (with $r = 7$)

• $\pi r^2 = 154$ (base area), $2\pi r = 44$ (circumference).
• CSA $= 44h$: $h = 5 \to 220$, $h = 10 \to 440$, $h = 14 \to 616$.
• $V = 154h$: $h = 10 \to 1540$, $h = 5 \to 770$.
• Radii 14, 21, 3.5 work the same way: $r = 14 \Rightarrow \pi r^2 = 616$; $r = 3.5 \Rightarrow \pi r^2 = 38.5$. Reverse questions hide in this family: volume 1540 with $r = 7$ is instantly $h = 10$.

Two worked mini-cases

Pipe: outer R $= 5$, inner r $= 3$, length 14 — ring $= \frac{22}{7}(25 - 9) = \frac{352}{7}$; volume $= \frac{352}{7} \times 14 = 704$ cu cm. The ($R^2 - r^2$) factorises to $(R+r)(R-r) = 8 \times 2$ — sometimes faster. Flow: a pipe of radius 2 cm delivers water at 10 cm/s for 60 s — volume $= \pi \times 4 \times 10 \times 60 = 2400\pi$ cu cm. Length delivered $=$ speed $\times$ time; the cross-section turns that length into volume.

Quick revision

• $V = \pi r^2 h$; CSA $= 2\pi rh$; TSA $= 2\pi r(h+r)$.
• $V \propto r^2 h$: radius changes square, height changes linear.
• Pipe metal $= \pi(R^2 - r^2)L$; litres $= 1000 \times \text{m}^3$.
• Melting/recasting: volume is invariant.