Step one. Calculate the proportional sizes of the Sun and moon during an annular eclipse. You'll find the Sun appears to be approximately 1.3 times the size of the moon, it will vary depending on which eclipse.

Step two. Create two isoceles triangles, one giving the angular distance of the Sun in the sky, one giving the angular distance of the moon. We can fill in these values.

We want the angles to be equal, to give the situation of a total eclipse. The angular distance of both objects will be the same. So in both triangles, the angle will be x. The side opposite this angle will have length M (the apparent size of the moon) in the moon triangle, and 1.3 M in the Sun triangle.

The remaining two sides will be k in the Sun triangle, and k+c in the moon triangle. k is the distance to the Sun. c is unknown, defined to just be the difference between the distance to the Sun and the distance to the moon. It will likely be negative.

Step three. Use the cosine rule to come up with a quadratic formula relating c and k.

Step four. Solve for c. If you don't know how to solve a quadratic equation, I can't help you.

I cannot write math into the forum, but I expect any capable reader will be able to run through and verify the calculations for themselves. Using p as the proportion (given as 1.3 above, but left open so that you may test with your own).

You will find that the distance to the moon, k+c, is k/p times the square root of (2-p^{2})

Certainly, there is some error in this calculation. According to RET the precise distances to the Sun and moon vary, as does p, but using p=1.3 and k as the distance to the Sun google gives me the distance to the moon as 64 million km. This is substantially different to the value RET gives.

For those interested in the theory behind this calculation, we begin by finding a proportion relating the Sun and moon. We then create, essentially, one large triangle. At one point is the observer, who looks up during a total eclipse to see the Sun and moon with the same angular size. They are different distances away however, so this triangle (currently a V, with the observer looking up) will have two lines opposite the angle, at varying distances away. One is the moon, the further is the Sun. Thus, there are two triangles in this one, the only differences being a) the size of the object, b) the distance to the object.

We can then use the proportion to relate the two distances, so b is the only unknown left to find. The distance to the Sun gives us the distance to the moon, and vice versa.

The RE values are dramatically far from what it is RET states.

**Edit**: Breakdown to demonstrate how many of the responses are either evasion or misunderstanding.

During a total eclipse, we can calculate the distance from the moon to the Earth in terms of their relative sizes, and the distance from the Sun to the Earth.

Their relative sizes stay the same during an annular eclipse. Only the distances change.

Calculate their relative sizes with pixel measurements of a photo during an annular eclipse, say, or whichever kind of measurement you prefer.

Use this knowledge to remove that unknown during a (limiting) total eclipse, where the Sun and moon appear the same size. You now have a gauge of distance to the moon in terms of distance to the Sun.

None of this relies on any assumptions, beyond the basic assumptions of math and logic. It applies to both FET and RET, but if we apply it to the RE numbers, they fail.