Question

A penny has a mass of 2.50 g, a diameter of 19.55 mm, and a thickness of 1.55 mm. Calculate the density of the material from which the penny is made.

Answer 1

Density = (mass) divided by (volume)

We know the mass (2.5 g).  We need to find the volume.

The penny is a very short cylinder.
The volume of a cylinder is (π · radius² · height).
The penny's radius is 1/2 of its diameter = 9.775 mm.
The 'height' of the cylinder is the penny's thickness = 1.55 mm.

Volume  =  (π) (9.775 mm)² (1.55 mm)

             =  (π) (95.55 mm²) (1.55 mm)

             =  (π) (148.1 mm³)

             =        465.3 mm³

We know the volume now.  So we could state the density of the penny,
but nobody will understand what we have.  Here it is:

         mass/volume = 2.5 g / 465.3 mm³  =  0.0054 g/mm³  .

Nobody every talks about density in units of ' gram/(millimeter)³ ' .
It's always ' gram / (centimeter)³ '.
So we have to convert our number for the volume.

                         (0.0054  g/mm³)  x  (10 mm / cm)³

                 =      (0.0054 x 1,000)  g/cm³

                 =          5.37  g/cm³  .

This isn't actually very close to what the US mint says for the density
of a penny, but it's in a much better ball park than 0.0054 was.