Today’s xkcd got me wondering how much land would we need if everyone stood six feet apart.

## Methodology

### We are all but spherical cows

If we want everyone to keep a minimum distance of $$r$$ from each other, the problem boils down to efficiently packing $$n$$ circles of radius $$r$$ in the smallest area possible. This can be calculated as follows:

Area of a circle with radius $$r$$,

$A_r = \pi r^2$

Optimal circle packing density1,

$\eta = \frac{\pi\sqrt{3}}{6}$

Area required for $$n$$ people,

\begin{align} A_n & = n \cdot \frac{A_r}{\eta} \\ & = n \cdot \frac{\pi r^2}{\frac{\pi \sqrt{3}}{6}} \\ & = 2 \sqrt{3} \cdot n \cdot r^2 \end{align}

### Crunching the numbers

Plugging in the current world population2 and the recommended distance of 6 feet3:

$n = \text{7,773,861,168}$ $r = 6\text{ft}$

we get

\begin{align} A_n & = 9.695×10^{11} \text{ft}^2 \\ & = 34,775 \text{mi}^2 \\ \end{align}

Dividing the area of San Francisco by $$2 \sqrt{3} \cdot 36 \text{ft}^2$$, gives us around 10.5 million people.