The shape of the atom and the nucleus is the same as the shape of the sphere, theoretically. Thus, the equation for their volumes are:
V = 4/3*πr³, where r is the radius
To be consistent with the units, let's use the SI unit meters. 1 fm = 10⁻¹⁵ m and 1 pm = 10⁻¹² m.
Volume of nucleus = 4/3*π(2.7×10⁻¹⁵)³ = 8.2×10⁻⁴⁴ m³
Volume of atom = 4/3*π(70×10⁻¹²)³ = 1.4×10⁻³⁰ m³
The percentage of nucleus in the atom is as follows:
8.2×10⁻⁴⁴ m³/1.4×10⁻³⁰ m³ * 100 = 5.9×10⁻¹² %
The nucleus of the atom occupies very little space in the atom, but is concentrated in mass. That's why the nucleus is very dense.
