To find out how many volumes fit in a larger volume, you obviously have to divide volume by volume, not radius. Likewise with areas.

Ok, so for a 10 nm diameter nanoparticle we have a radius of 5 nm, and the covalent atom radius is 0.145, and the volume is 524 nm

^{3} then, the percentage of atoms on the surface must have the same ratio as the volume of thickness of atoms on the surface to entire volume of the nanoparticle, like…

O = Outer radius of the nanoparticle (outside of the atoms on the surface) = 5 nm

I = Inner radius of the nanoparticle (inside of the atoms on the surface) = 5 – 0.145 = 4.855 nm, so…

Volume of the entire nanoparticle = 524 nm3

Volume of the “inner” part of the nanoparticle (excluding 1 layer of atoms) =

= ( 4 x Pi x I

^{3} ) / 3 = 479 nm3

So the outer layer of atoms on the surface have a volume of 45 nm3 (524 – 479)

Surface ratio = 45 / 524 = 8.6%

So less than 9% of the atoms are in fact on the surface of a 10 nm diameter spherical silver nanoparticle.