Question

An equilateral triangle with sides of length $s$ is rotated in space about one side.

Show that the volume of the solid formed is $\frac{1}{4}\pi s^3$ .

Answer 1

Explanation:

Use Pappus theorem:
$V$ = volume generated
$A$ = surface area generated
$A_t$ = generating area
$L$= generating length of curve
$C$ = distance traveled by the centroid

$h=\sqrt{s^2-(\frac{1}{2}s{)}^2}=\frac{\sqrt{3}}{2}s$

$\overline{y}=\frac{1}{3}h=\frac{1}{3}(\frac{\sqrt{3}}{2}s)=\frac{\sqrt{3}}{6}s$ ← centroid of area

$A_t=\frac{1}{2}sh=\frac{1}{2}s(\frac{\sqrt{3}}{2}s)=\frac{\sqrt{3}}{4}{s}^2$

$V=A_t\times C=A_t\times 2\pi \overline{y}$

$V=\frac{\sqrt{3}}{4}{s}^2\times 2\pi (\frac{\sqrt{3}}{6}s)$

$V=\frac{1}{4}\pi s^3$

Therefore,$V=\frac{1}{4}\pi s^3$ is the required answer.