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During a lunar or solar eclipse, from our perspective when we look towards the moon and the sun is behind the moon, does it fit exactly?

Has anyone calculated the sizes of each in a relative context? Obviously the sun is thousands of times larger, but I am talking about the size of the sun/moon when looking from our perspective.

I hope my question makes sense.

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    $\begingroup$ Short comment due to lack of time. Since both distances, moon-earth and sun-earth, vary periodically, so do the apparent sizes of moon and sun. The moon can be smaller or larger than the sun; see 'annular eclipse' for example. How much larger the moon appears determines (among others) the length of the totality phase of the eclipse. $\endgroup$ – taupunkt Sep 15 '14 at 15:26
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To expand on what taupunkt commented, on average, according to the article Solar Eclipse: What is a Total Solar Eclipse & When is the Next One? (Rao, 2014), it is a unique quirk of nature that

The sun's 864,000-mile diameter is fully 400 times greater than that of our puny moon, which measures just 2,160 miles. But the moon also happens to be about 400 times closer to the sun than the Earth

This is often stated in middle and high school science textbooks.

However, the orbits of the Earth around the sun and the moon around the Earth are elliptical, as illustrated in the diagram below:

enter image description here

Image source: NOAA

This results in the exact ratio mentioned above not occurring resulting in an annular eclipse, where the moon is too 'small' to fully cover the solar disc.

A comparison between a total and annular eclipse is shown below:

enter image description here

Image source: ScienceJedi

NASA's Eclipse data page, in particular, their Eclipses and moon orbit information page discuss in detail (with considerable data), the

interaction and harmonics of the synodic, anomalistic, and draconic months not only determine how frequently eclipses occur, but they also control the geometric characteristics and classification of each eclipse.

Within the NASA link, data about how much of the sun's disk that would be obscured by the moon has been calculated (from observations and predicted). For example, using the data for solar eclipses from 2011 to 2020, The average fraction of the sun's disk that is and is expected to be obscured (this is referred to as 'Eclipse magnitude') in an annular eclipse varies from about 0.95-0.99 (95% to 99%) for that time period.

A catalog of eclipses up to the year 3000 is provide on this NASA page.

The greatest difference would occur if the moon is at its furthest from the Earth (apogee) and the Earth closest to the sun (perihelion).

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