## Standing Room Only

In 2012 the global fertility rate - births per woman - was 2.5, compared to 2.0 needed for a stable population, assuming no stillbirths, and that all babies born survived to have children themselves. This means that every couple produced 2.5 children and over the course of a generation the population increases by a factor of
$\frac{2.5}{2.0}=1.25$
.
In 2012 the world population was about 7 billion. After n generations the world population would be
$7 \times 10^9 \times 1.25^n$
.
the Earth is roughly a sphere of
$6370 km = 6.370 \times 10^6 m$
so has a surface area equal to
$4 \pi r^2 = 4 \pi \times (6370 \times 10^6)^2 =5.1 \times 10^14 m^2$

Over the whole surface of the Earth, each person will have only 1 square metre to call his own when the population equals the surface area.
$7 \times 10^9 \times 1.25^n =5.1 \times 10^14$

This will be after
$n = \frac{ln((5,1 \times 10^14)/(7 \times 10^9))}{ln 1.25} = 50$
generations.
If we take a generation as 45 years, there will be standing room only after
$50 \times 45 = 2250$
years.