# Hollow Circular Cylinder

Some examples of a hollow cylinder are pipes, circular buildings and bearing bushes. If $R$ is the outside radius of the cylinder and $r$ is the inside radius of the cylinder, then (i)         $V = \pi {R^2}h – \pi {r^2}h = \pi \left( {{R^2} – {r^2}} \right)h$
(ii)        $V = \frac{\pi }{4}\left( {{D^2} – {d^2}} \right)$

with $D$ and $d$ being the outer and inner diameter, and $V$ is volume.

Example:

A well with a 10m inside diameter is dug 14m deep. The earth taken out of it is spread all around to a width of 5m to form an embankment. Find the height of the embankment.

Solution:
Volume of the dug out earth          $= \pi {r^2}h$
$= \frac{{22}}{7} \times 5 \times 5 \times 14$ $1100\,{\text{cu}}{\text{.}}\,{\text{m}}$

$= \pi \left( {{R^2} – {r^2}} \right)$
$= \pi \left( {{{10}^2} – {5^2}} \right)$           $= 75 \times \frac{{22}}{7}\,{\text{sq}}{\text{.m}}$

$\therefore$            Height of the embankment

$= \frac{{{\text{Volume of the earth dug out}}}}{{{\text{Area of the embankment}}}}$

$= \frac{{1100}}{{75 \times \frac{{22}}{7}}}\,\, = \frac{{14}}{3}\,\,\, = 4\frac{2}{3}\,\,{\text{m}}$

Example:

A hollow cylinder copper pipe is 21dm long. Its outer and inner diameters are 10cm and 6cm respectively. Find the volume of the copper used in making the pipe.

Solution:

Given that:
The height of the cylindrical pipe is $h = 21{\text{dm}}\,\,\, = \,210{\text{cm}}$

$\therefore$            External radius, $R = \frac{{10}}{2}\,\, = \,5{\text{cm}}$
Internal radius, $R = \frac{6}{2}\,\, = \,3{\text{cm}}$
Volume of the copper used in making the pipe
$= {\text{ Volume of external cylinder }} – {\text{ volume of internal cylinder}}$
$= \pi {R^2}h – \pi {r^2}h\,\,\,\,\,\, = \,\pi \left( {{R^2} – {r^2}} \right)h\,\,\,\, = \frac{{22}}{7}\left[ {{5^2} – {3^2}} \right] \times 210$
$= \frac{{22}}{7} \times 16 \times 210\,\,\,\, = 22 \times 16 \times 30\,\,\,\, = 10560\,{\text{cu}}{\text{.cm}}$