Which is bigger: the Sun or the Moon?

Of course you know the answer to this. (Though when I ask parents the question, they tend to look sheepish and say nothing, for fear of being wrong!)

Actually, it's not at all obvious which one is bigger: after all, they look the same size. You know which one is bigger because you were told it when you were very young. Which rather spoils the pleasure of trying to work it out. But how do we know?

It's fairly easy to establish that the Sun is a least a bit bigger than the Moon. They look the same size in the sky, but during a solar eclipse the Moon completely covers the face of the Sun. So the Sun must be further away than the Moon and therefore it must be bigger.

To establish how much bigger it is we need another rare sighting in the sky: have you ever seen the Moon and the Sun in the sky at the same time? It does happen (Lewis Carroll wrote about it in the first two verses of The Walrus and the Carpenter.) Well, imagine that you can see them both in the sky together but that the Moon is a half Moon.

What must the geometry of the solar system be at that moment? (There's a clue in the picture above, which you can enlarge by clicking on it.) The Sun-Moon-Earth forms a giant right-angled triangle, with the Moon at the right-angle. If we can measure the Sun-Earth-Moon angle, then we can use trigonometry to find the relative distances of the Sun and the Moon from Earth. Since the Sun and the Moon look the same size in the sky, these relative distances must also be the relative sizes of the Sun and Moon themselves.

The hard part is measuring the angle. You need it to be exactly a half Moon. You need to measure the angle between the Sun (without looking at it directly), you and the Moon.

But it gets worse, if you're even slightly wrong with your measurement, the relative distances and sizes change by a surprisingly large amount. This is because the angle is nearly 90° – and that is because the Sun is really very much farther away than the Moon and very much bigger. In fact the angle is about 89 5/6°, which indicates that the Sun is about 340 times the size of the Moon. But if you measure it as 89 4/6°, you'll get the Sun being only 170 times the size of the Moon, in other words half the size! So your measurement needs to be incredibly precise!

On question remains: now that we know the relative sizes of the Sun and the Moon, how do we know their actual sizes?

Update on Tuesday, March 23, 2010 at 11:39AM by Matthew Handy

A few notes.

The diagram above is merely intended to indicate the positions of the Sun, Moon and Earth. The sizes are not to the same scale (compared with the Earth, the Sun should be much bigger and the Moon slightly smaller) and the distances are not in proportion (the Earth-Sun distance is much bigger than the Earth-Moon distance – indeed, the triangle should look almost like two parallel lines).

The Sun is, in fact, almost exactly 400 times the size of the Moon, by which I mean its radius is about 400 times the size of the the radius of the Moon. This means that its volume is 400x400x400 times the volume of the Moon, which makes it nearly three quarters of a million times as large. (The ratio of the masses of the Sun and the Moon is different, however, because they do not have the same average density.)

The figures of 340 and 170 come from tan(89 5/6°) and tan(89 4/6°). It's an incredible coincidence that, although the angle differs by one-sixth of a degree, one tangent is (almost exactly) double the size of the other. (In fact, it's double to one part in a hundred thousand!) Why does a small change in the angle result in such a big change in the tangent of it?