Problem — How many hairs on a head

Problem

Estimate the order of magnitude of the number of hairs on a healthy adult head, reasoning from the surface density of hairs (a few per $\text{mm}^2$) and the area of the scalp. Compare your result with the tabulated value ($\sim 10^5$ total hairs) and indicate the most critical assumptions.

Solution

Step 1 — Surface density. The statement suggests “a few hairs per $\text{mm}^2$”. We cautiously assume about $2\;\text{hairs/mm}^2$. Since $1\;\text{cm}^2 = 100\;\text{mm}^2$:

$\sigma \approx 2\;\frac{\text{hairs}}{\text{mm}^2} = 2\cdot 100 = 200\;\frac{\text{hairs}}{\text{cm}^2}$

Step 2 — Scalp area. We approximate the hair-covered area to a cap of the head. With an effective radius of $\sim 8\;\text{cm}$ we get an area of the order of:

$A \approx 600{-}700\;\text{cm}^2 \approx 6\cdot 10^2\;\text{cm}^2$

Step 3 — Total number.

$N = \sigma\,A \approx 200 \cdot 6\cdot 10^2 = 1{,}2\cdot 10^5 \approx 10^5$

Result.

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

Step 4 — Comparison and critical assumptions. The tabulated value ($\sim 10^5$ hairs) matches the estimate as an order of magnitude. The most critical assumptions are:

• the density $\sigma$, which varies greatly between individuals and areas of the head (higher at the crown, sparser towards the edges): a factor of $2$–$3$ uncertainty;
• the area $A$, which depends on the hairline and the extent of the hair-covered area.

Since the result is the product of two factors each known to within a factor $\sim 2$, the exponent $10^5$ is robust even though the leading digit (between $0{,}5$ and $2\cdot 10^5$) is uncertain. This is the typical outcome of a Fermi estimate: the power of $10$ is reliable, the coefficient is not.

Linked notes

Topics: Grandezze fisiche e misura Concepts: Notazione scientifica e ordini di grandezza Skills: Stime di Fermi