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The hydrogen atom contains 1 proton and 1 electron. The radius of the proton is approximately 1.0 fm (femtometers), and the radius of the hydrogen atom is approximately 53 pm (picometers).

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Do you mean, how do you find the volume from the given radii, or how these radii are obtained? The former is straightforward (see answer below), the latter is a bit more complicated. –  Benji Remez Oct 20 '12 at 22:05
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2 Answers

The Size of a proton is an extremely difficult calculation, It cannot be done by hand(so far). It requires an in depth understanding of Strong forces(Non-Abelian Gauge theory) and super-computers(See Lattice QCD)

The radius of the Hydrogen atom is relatively straight forward.(See Bohr Radius) On can derive it from old quantum theory thought angular-momentum Quantization.

you will find the derivations here.

http://en.wikipedia.org/wiki/Bohr_model

http://en.wikipedia.org/wiki/Bohr_radius

Radius to volume is easy

$$V=\frac{4}{3}\pi r3$$

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Indeed the proton calculation is so hard that the right thing to do is measure the size. But even then there is some ambiguity concerning what you mean by radius. You can choose the RMS charge radius, half the FWHM charge radius, the strong nuclear interaction hard-core repulsion radius, or various figures taken from the parton distribution functions in the forward scattering limit. Most of the choices fall somewhere between 0.5 and 1.5 fm, and it is common to see people pick 0.9 or 1.0 fm to use for BoTE calculations. –  dmckee Oct 20 '12 at 22:41
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Model them as spheres. The volume of a sphere is $$V = \frac{4}{3}\pi r^3.$$ This should straightforwardly give you the answer.

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What's the old joke? First, assume the sheep is a sphere ... –  McGarnagle Oct 21 '12 at 18:28
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