Measure the angle between the Moon and
the Sun when the Moon is exactly in its first quarter phase. This angle will be almost, but not quite 90.0 degrees.
The Sun-Moon-Earth angle **is** exactly 90 degrees,
so the Sun, Moon and Earth form a very skinny right triangle at the
instant of the Moon's first (or third) quarter phase. We know two of the
triangle's interior angles (90 degrees, and not quite 90 degrees), and the
length of the enclosed side (384,400 km),
so we can use trigonometry to
determine the distance between the Earth and Sun.

Of course, now that we have RADAR, we don't have to be so clever.

Average distance to the Sun (a.k.a. one Astronomical Unit): 150 Million km.