I think that this is more or less the approach I would take, but you shouldn't worry about the actual diameter of anything. It's not important, after all - if everything was scaled up twice as big, the answer would be the same. Just call the diameter of the cup a nice round number and then see how the hazelnuts compare to it. In this case I think there's about five hazelnut widths to the glass, so I'm gonna call the glass diameter 50, the nuts 10, and the glass height 80.

You'll need to change your formulae, though. pi*d is the circumference of a circle, but we need the area here, so pi*r*r (and then multiply by height for volume). That gives me 157,050 whateverunits cubed for the volume of the cup. For a sphere it's (4/3)*pi*r*r*r, so 524 for the hazelnuts. Now, I know that spheres don't pack perfectly into a volume, but I don't remember the factor even for optimal packing, so I'm just gonna take a wild guess and say that 70% of the internal volume of the cup is actually occupied by hazelnuts. That gives me... 209 hazelnuts in the cup. Which seems worse than your answer on a gut level, but I can count 86 visible ones so it's maybe actually not bad