A team of surveyors are given the task of finding the size of an equilateral triangular plot.

The remarkable thing is that the condition that the triangle is
equilateral is not even needed!

The area of a spherical triangle of angles A,B,C is proportional to the
difference between A+B+C and 180 degrees.  [This can actually be proved
without much difficulty by extending the sides to great circles and
using wherever possible the fact that the area between two meridians is
proportional to the angle between them at the poles.]

Now a spherical triangle all of who angles are 90 degrees covers exactly
1/8 of the surface area of the sphere.  So, if A+B+C-180 = .01 second =
1/360000 degree then the triangle's area is the surface area of the
sphere divided by 720*360000 = 259200000 [Editor: was 25920000].

Now the surface area of a sphere of radius r is 4*pi*r^2; but we still
don't know what r is.  However, since the problem is "theoretical",
let's assume that the circumference of the earth is exactly 40000
kilometers -- that's how the kilometer was meant to be defined, though
it turns out that the measurement was off by about %.01 and the
slightly too short kilometer became standard.  (Anyway if we want to be
that accurate we also run into the problem that the earth isn't exactly
spherical, so the choice of spherical approximation is more or less up
to us.)  So, 2*pi*r = 40000 km, whence r = (20000/pi) km, and the
surface area of the earth is (40000^2/pi) km^2.  Dividing this by
259200000 yields (500/81pi) km^2, which numerically comes to
1.96487584... square kilometers, or just over 3/4 of a square mile --
barring silly arithmetical errors.

-- Noam D. Elkies <elkies@fuxpam.math.harvard.edu>