Problem — Hairs on a head

Problem

Estimate the number of hairs present on a human head. Explain which data you assumed (area of the scalp, density of hairs per $\text{cm}^2$) and express the result as an order of magnitude.

Solution

Step 1 — Area of the scalp. We model the hairy part of the head as a cap of radius $\sim 8\;\text{cm}$. Its area is of the order of a few hundred $\text{cm}^2$; we assume a reasonable value:

$A \approx 600\;\text{cm}^2$

Step 2 — Density of hairs. Observing the scalp one counts on the order of one or two hairs per $\text{mm}^2$, i.e. about a hundred per $\text{cm}^2$. We assume:

$\sigma \approx 150\;\text{hairs/cm}^2$

Step 3 — Total number. The number of hairs is the product of area and density:

$N = \sigma\,A \approx 150\cdot 600 = 9\cdot 10^4 \approx 10^5$

Result.

$\ev{N \sim 10^5\;\text{hairs}}$

Discussion. The two assumptions (area and density) are each known to within a factor of $2$–$3$; their product is therefore reliable as an order of magnitude. The tabulated value for a human head ($\sim 10^5$ hairs) confirms the estimate.

Linked atoms

Argomenti: Physical quantities and measurement Concetti: Scientific notation and orders of magnitude Competenze: Fermi estimates