From Wikipedia: Fermi was known for his ability to make good approximate calculations with little or no actual data, hence the name. One example is his estimate of the strength of the atomic bomb detonated at the Trinity test, based on the distance travelled by pieces of paper dropped from his hand during the blast. Fermi's estimate of 10 kilotons of TNT was remarkably close to the now-accepted value of around 20 kilotons, a difference of less than one order of magnitude.
I have not been able to find any references explaining how he made this calculation. Please provide a reference or an example calculation.
See also: http://en.wikipedia.org/wiki/Fermi_problem
Answer
I describe this in my book "Guesstimation 2.0" (Princeton University Press, 2012). The work done by the expanding shock wave is pressure times change in volume. The change in volume is as described by the previous answer: $2.5\,\mathrm{m} \times 2\pi \times (16\,\mathrm{km})^2$.
Fermi could feel the extra pressure due to the explosion, we can only estimate it. The extra force on his body must have been more than 100 N (20 lb) and less than $10^4\,\mathrm{N}$ (2000 lb) so we will take the geometric mean and estimate $10^3\,\mathrm{N}$. The extra pressure then is just $10^3\,\mathrm{N}$ divided by the typical frontal surface area of a person of $1\,\mathrm{m}^2$ or $P = 10^3\,\mathrm{N}/\mathrm{m}^2$.
Then $E = P \Delta V = (10^3\,\mathrm{N}/\mathrm{m}^2)(4 \times 10^9\,\mathrm{m}^3) = 4 \times 10^{12}\,\mathrm{J} = 1\,\mathrm{kT}$
Now multiply by a few because the energy of the bomb can go into light (photons), nuclear radiation, shock wave, ground pulse … and we get an estimate of 4 kT.
Fermi was there and he estimated 10 kT so this is in the correct ballpark.
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