*When there is an eclipse, the moon blocks out almost the entire sun relative to what we see here on earth because they take up the same earth-observed degree size. Why does the moon, have the same exact observable degree size as the sun? Is it just a coincidence? What are the chances that this should be? Follow up question - are there any other planetary moons which also project a disc on their planet's surface which exactly matches the sun as observed from that planet? Do any come close either way at all?*

As you say - it's just a coincidence that the Moon and the Sun have such similar angular sizes as seen from Earth. In fact this won't last very long (on astronomical timescales). The Moon is moving away from the Earth (albeit very slowly), so eventually will have a smaller angular size than the Sun and we will no longer have total solar eclipses.

The figure out if any other planets can also have total solar eclipses is actually very easy to work out for yourself. I'll tell you how by giving you a couple of examples and then leave you to it!

If you go to (eg.) the Nine Planets site you can find the size and orbit of all the moons of all the planets. For example:

The Moon

orbit: 384,400 km from Earth

diameter: 3476 km

We also need to know the size of the Sun and how far it is from the planet in question so

The Earth

orbit: 149,600,000 km (1.00 AU) from Sun

The Sun

diameter: 1,390,000 km

The angular size of an object is (roughly) its diameter divided by how far away it is (this gives the angular size in radians if both are in the same units) so...

The Sun

angular size 1,390,000km/149,600,000km = 0.009 radians

= 0.5 degrees

(A useful number is that 1 radian = 57 degrees, so I multiply the size in radians by 57 to get the size in degrees which I understand better).

The Moon

angular size = 3476km/384,400km = 0.009 radians

= 0.5 degrees

They are the same as you already knew.

Let's try one more example. At random I'm going to choose Titan, the largest moon of Saturn:

Titan

orbit: 1,221,830 km from Saturn

diameter: 5150 km

I also need to know that Saturn is orbiting at a distance of 1,429,400,000 km (9.54 AU) from the Sun.

The Sun

angular size from Saturn = 1,390,000km/1,429,400,000km = 0.0010 radians

= 0.06 degrees.

(an easier way to do this is to say that Saturn is 9.5 times further from the Sun than the Earth is so the Sun has an angular size 9.5 times smaller).

Titan

angular size from Saturn

= 5150km/1,221,830km = 0.004 radians

= 0.2 degrees.

So Titan and the Sun do not have approximately the same angular size as seen from Saturn. Since the Sun appears considerably smaller than Titan, Titan can cast a shadow on (part of) Saturn. Thus, a hypothetical observer on Saturn could occasionally see total solar eclipses due to Titan passing in front of the Sun.

Have fun finding if any of the moons do match!

*Update:* NASA's Cassini spacecraft observed Titan's shadow on Saturn in 2009. Check out the image at http://photojournal.jpl.nasa.gov/catalog/PIA11648.

*This page was last updated June 28, 2015.*