Question

Estimate the mass of human head.
Use Fermi calculations.

Answer 1

The mass is given as follows:

$$m = \rho V$$

where $\rho$ is the density of a head, and $V$ is its volume. Let's approximate the head as a perfect shere. Then its volume is approximately 4 times its radius cubed or 0.5 times diameter cubed (assuming $\pi\approx 3$ ):

$$V \approx 0.5d^3$$

Let's estimate the diameter of the head as $d\approx 20cm = 0.2m$. Also, since the head consists mostly of brain (which has almost the same density as water) and bones (which are a little more dense than water), let's take the average density to be $\rho \approx 1200 kg/m^3$.

Finally, obtain:

$$m\approx 0.5d^3\rho\approx 0.5\cdot (0.2m)^3\cdot 1200kg/m^3 \approx 4.8kg$$

The actual masses varies from approximately 4kg to 6 kg (see, for example, https://www.sciencedirect.com/science/article/abs/pii/S0021929009001444), thus, the obtainded estimation is quite good.

Answer. 4.8 kg.