# What's the smallest country that can fit everyone standing six feet apart?

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 density^{1},

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

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 population^{2} and the recommended distance of 6 feet^{3}:

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

$r = 6\text{ft}$

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 Wikipedia^{4}, this is just shy of the land area of **Portugal 🇵🇹**.
We could also squeeze into **Hungary 🇭🇺** if some of us were willing to swim.

## #Corollaries

### #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
world^{5}.

Shrey Banga

banga.shrey@gmail.com