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


hollow-circular-cylinder

            (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


hollow-circular-cylinder-1

1100\,{\text{cu}}{\text{.}}\,{\text{m}}

            Area of the embankment (shaded)

                                                             = \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}}