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


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

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

And the answer is…

According to Wikipedia4, this is just shy of the land area of Portugal 🇵🇹. We could also squeeze into Hungary 🇭🇺 if some of us were willing to swim.


How many people can San Francisco fit standing 6 feet apart?

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

Are there any countries where people couldn’t stand 6 feet apart?

1 square mile of land can have 223,563 people standing 6 feet apart. This is much higher than the most densely populated countries and territories in the world5.