the sun - How do you define the diameter of the Sun

I thought I'd contribute an answer because there's a very recent paper on the subject:

Measuring the solar radius from space during the 2012 Venus Transit

It appeared in my RSS feeds this morning! A related writeup is online at the HMI website.

To answer the question, this measurement uses the transit of Venus to fit the limb-darkening law of the Sun. That is, the Sun is a bit fainter the further from the centre that you look. As you reach the optically thinner layers near the "surface", the brightness falls off rapidly, towards zero in the vacuum of space. The inflection point of the curve (as a function of distance from the centre of the disk) is a reasonable estimate of the "radius". As pointed out elsewhere, the value changes depending on which wavelength you use, but only by a few hundred km, compared to the Sun's overall radius of about 700 000km (actually more like 695 946 km), so the uncertainty is at or below the 0.1% level. Phil Plait wrote about a similar measurement (by the same team, I believe) that used the transits of Mercury in 2003 and 2006.

Finally, the team also used the limb-darkening (I think) to measure how round the Sun is. i.e. the diameter from top-to-bottom versus left-to-right. Answer: the Sun is very very round, with the radii differing by a few parts per million.