You can see why the usual illustrations aren't to scale. Incidentally, the Sun would be hundreds of screen widths away.
Okay, here's where it gets serious. The picture of the eclipse about to happen is drawn just with lines.
Inside this picture are three triangles struggling to get out
The three triangles, imaginatively named A, B and C so you can tell this is real mathematics, are all the same shape, but different sizes. In geometric jargon, this is what is meant by similar triangles. This means that if we know that one triangle is, say, twice as big as another, we can tell that corresponding sides of the two triangles are also twice as big. (B and C are obviously the same shape, and A has the same small angle as C, because of what we said earlier about the sun and moon being the same visual size: all the triangles are 110 times longer than wide)
So What, I hear you say.
Well, the length of triangle A is what we want to find: the distance to the moon. We've just measured the eclipse shadow to be 2.7 times the moon size, and we note with interest that these two things are the corresponding (smallest) sides of triangles A and B! So B is 2.7 times the size of A. But lo, the length of triangle C is just the sum of lengths of A and B, so C is 1 + 2.7 = 3.7 times the size of A.
We're almost there. We mentioned earlier a way to get at the length of C: the sun's shadow is 110 times longer than the width of the shadow caster, so the length of C is 110 earth diameters. This is 3.7 moon distances (see previous paragraph you skipped)
That's right! if we divide, 110/3.7 = 30, and we have determined that the moon is 30 earth diameters away! (that's why they paid the Greeks the big bucks)
We can now easily see the relative sizes of the earth and moon. Look again at the similar triangles A and C: one set of corresponding sides is the diameters of the earth and moon, which are in the same ratio as the size of the triangles, or 3.7 to 1, so the Earth's diameter is about 3.7 times the moon's diameter.
Finally then, what do the earth and moon really look like, when drawn correctly?
You can see why the usual illustrations aren't to scale. Incidentally, the Sun would be
hundreds of screen widths away.
What else can we do with this?
Next:
Further Adventures