Mensuration (3D)

The sphere family

For a sphere of radius $r$: $V = \frac{4}{3}\pi r^3,\qquad \text{SA} = 4\pi r^2$ For a hemisphere of the same radius: curved surface $= 2\pi r^2$ (half the sphere's), flat face $= \pi r^2$, and total $= 3\pi r^2$ — the famous "3" that examiners love. Volume of a hemisphere $= \frac{2}{3}\pi r^3$. The radius is the only input: every question is "get $r$, then apply one formula". Given a diameter? Halve it first. Given the surface area? $r = \sqrt{\text{SA}/4\pi}$. Given the volume? $r = \sqrt[3]{\frac{3V}{4\pi}}$.

Melting, scaling and submersion

• Melting preserves volume: a big sphere recast into $n$ small equal spheres gives $r_{\text{small}} = \frac{r}{\sqrt[3]{n}}$ — for $n = 8$ the radius halves, for $n = 27$ it divides by 3.
• Scaling: radius $\times k$ → surface $\times k^2$, volume $\times k^3$. "Radius doubled" means SA $\times 4$ and $V \times 8$; "radius halved" gives $\frac14$ and $\frac18$.
• Submersion: a solid dropped into a (partly filled) cylinder of base radius $R$ raises the water level by $h = \frac{V_{\text{solid}}}{\pi R^2}$ — the displaced volume spreads over the base.

Same-base relations

Sphere vs cylinder of radius $r$, height $2r$ (the sphere fits exactly inside): both have volume $\frac{4}{3}\pi r^3$. Sphere vs hemisphere: hemisphere is half the volume, $\frac34$ of the total surface ($3\pi r^2$ vs $4\pi r^2$).

The 22/7 number family

• $r = 7$: SA $= 616$, $V = \frac{4}{3} \times \frac{22}{7} \times 343 = 4312/3$ — exams avoid it; they use $r = 21$: $r^3 = 9261$, $V = 4 \times 22 \times 441 = 38808$, SA $= 4 \times \frac{22}{7} \times 441 = 5544$.
• Hemisphere at $r = 7$: CSA $308$, TSA $462$; at $r = 21$: CSA $2772$, TSA $4158$.
• The displacement pair worth memorising: a sphere of radius 3 displaces $36\pi$; a cylinder of radius 4 therefore rises $\frac{36}{16} = 2.25$ cm.

Scaling in percent form

Radius $+10\%$: SA $+21\%$, $V$ $+33.1\%$ (from $1.1^2$ and $1.1^3$). Radius $-50\%$: SA $\frac14$, $V$ $\frac18$ — volume swings harder than area in both directions. Percent-change questions in this subtopic are always these two exponent rules; if you find yourself expanding $\frac{4}{3}\pi$, stop and write $k^2$ or $k^3$ instead.

Quick revision

• $V = \frac43\pi r^3$; SA $= 4\pi r^2$; hemisphere: CSA $2\pi r^2$, TSA $3\pi r^2$, $V = \frac23\pi r^3$.
• Radius $\times k$: SA $\times k^2$, $V \times k^3$.
• Melt into $n$ equal spheres: radius $\div \sqrt[3]{n}$.
• Water rise: $h = \frac{V_{\text{solid}}}{\pi R^2}$ of the container.