A sphere has no flat base to stack, so the cylinder trick needs a twist. More than 2,000 years ago, Archimedes compared a sphere with a cylinder and a cone that just fit around it.
Por qué funciona
A sphere of radius $r$ fits snugly in a cylinder of radius $r$ and height $2r$. That cylinder holds $\pi r^{2} \cdot 2r = 2\pi r^{3}$, and a cone with the same base and height holds one third as much, $\frac{2}{3}\pi r^{3}$. Archimedes showed that the cone and the sphere together hold exactly as much as the cylinder. So the sphere holds $2\pi r^{3} - \frac{2}{3}\pi r^{3} = \frac{4}{3}\pi r^{3}$: the cone, sphere and cylinder are in the ratio $1 : 2 : 3$.
Idea clave
$$V = \frac{4}{3}\pi r^{3}$$ Cube the radius (not the diameter), then multiply by $\frac{4}{3}\pi$.
Worked examples
Ejemplo resuelto: From the radius
Find the volume of a sphere with radius 3 cm.
- $r^{3} = 27$.
- $V = \frac{4}{3}\pi \cdot 27 = 36\pi \approx 113$.
Respuesta: $36\pi \approx 113$ cm³
Ejemplo resuelto: From the diameter
A ball has diameter 12 inches. Find its volume.
- $r = 6$, so $r^{3} = 216$.
- $V = \frac{4}{3}\pi \cdot 216 = 288\pi \approx 904$.
Respuesta: $288\pi \approx 904$ cubic inches
Ejemplo resuelto: Half a sphere
A bowl is a hemisphere with radius 6 cm. How much can it hold?
- A whole sphere: $\frac{4}{3}\pi \cdot 216 = 288\pi$.
- Half of it: $144\pi \approx 452$.
Respuesta: $144\pi \approx 452$ cm³
Ejemplo resuelto: Work backward
A sphere has volume $36\pi$. Find its radius.
- $\frac{4}{3}\pi r^{3} = 36\pi$. Divide by $\pi$ and multiply by $\frac{3}{4}$: $r^{3} = 27$.
- $r = \sqrt[3]{27} = 3$.
Respuesta: $3$
Error común
Squaring the radius instead of cubing it, or cubing the diameter. Volume is three-dimensional, so the radius appears three times. With the diameter 12, cubing 12 instead of 6 makes the answer 8 times too big.