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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.


We are all but spherical cows

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

Area of a circle with radius rr,

Ar=πr2A_r = \pi r^2

Optimal circle packing density1,

η=π36\eta = \frac{\pi\sqrt{3}}{6}

Area required for nn people,

An=nArη=nπr2π36=23nr2\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=7,773,861,168n = \text{7,773,861,168}

r=6ftr = 6\text{ft}

we get

An=9.695×1011ft2=34,775mi2\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 2336ft22 \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

Shrey Banga