Volume of a Hollow Cylinder (Formula & Derivation)

A hollow cylinder is essentially a thick pipe. It has an empty hole running through the center. Therefore, it has two different circles at its base:

1. An Outer Circle with an Outer Radius (denoted by capital $R$).
2. An Inner Circle (the empty hole) with an Inner Radius (denoted by small $r$).

• The height (or length) of the pipe is denoted by $h$.

To find the volume of the actual material (like the metal or plastic) making up the pipe, you use a simple subtraction logic:

• First, pretend the cylinder is completely solid and calculate its Total Outer Volume.
• Then, calculate the Volume of the Empty Hole inside it.
• Finally, subtract the empty hole from the total volume to get the volume of the metal.

Volume of Material = Outer Volume - Inner Volume

We know the volume of a standard cylinder is $\pi r^2 h$. Applying our logic:

• Outer Volume = $\pi R^2 h$
• Inner Volume = $\pi r^2 h$

Volume of Hollow Cylinder = $\pi R^2 h - \pi r^2 h$

By taking $\pi$ and $h$ common, we get the final, simplified formula: $V = \pi h (R^2 - r^2)$