What are the units of volume? Suppose another student tells you that a cylinder of radius r and height h has volume given by ${\mathbf{\pi r}}^{\mathbf{3}}\mathbf{h}$. Explain why this cannot be right.

Volume is scalar quantity expressing the size of a three dimensional space enclosed by a closed surface. For example, the space that a substance or 3D space occupies or contains volume is often quantified numerically using the SI- derived unit, the cubic meter.

The unit of volume in the CGS system is cubic centimeters $\left({\mathrm{cm}}^3\right)$.

The unit of volume is the S.I system is cubic meters $\left({\mathrm{m}}^3\right)$.

Given that volume of the cylinder is,

$V={\mathrm{\pi r}}^3\mathrm{h}$

Here, r is the radius of the cylinder in meter and h is the height of the cylinder in meter.

The dimensional formula for given cylinder is,

$\begin{array}{rcl}\mathrm{V}& =& \left[{\mathrm{m}}^3\right]\left[\mathrm{m}\right]\\ & =& \left[{\mathrm{m}}^4\right]\end{array}$

Hence, the unit for volume of cylinder is ${\mathrm{m}}^4$.

But the correct unit for volume of cylinder is ${\mathrm{m}}^3$.

Therefore, the given volume of cylinder is not right.

In CGS system, the unit for radius is centimeter $\left(\mathrm{cm}\right)$, and the unit of height is centimeter $\left(\mathrm{cm}\right)$.

The dimensional formula for given cylinder is,

$\begin{array}{rcl}\mathrm{V}& =& \left[{\mathrm{cm}}^3\right]\left[\mathrm{cm}\right]\\ & =& \left[{\mathrm{cm}}^4\right]\end{array}$

Therefore, the unit for volume of cylinder is ${\mathrm{cm}}^4$. But the correct unit for volume of cylinder is $\left({\mathrm{cm}}^3\right)$.

Hence, the given volume of cylinder is not right.