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 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.

#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 world⁵.

1. Circle packing - Wikipedia ↩

2. Current World Population - Worldometers ↩

3. Prevention of Coronavirus Disease 2019 (COVID-19) - CDC ↩

4. List of countries and dependencies by area - Wikipedia ↩

5. List of countries and dependencies by population density - Wikipedia ↩