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The fool that I am, today was the first time I wondered how come the shadow of the earth is almost same in size as the moon. I mean, as I was looking at the crescent moon in the evening, it came upon me that the crescent was like this:

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and not like this:

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neither do we see as the moon like this:

enter image description here

Of course, this must be because the sizes of moon and earth along with the distance between them. However my question is if this is a mere coincidence, one of the many things that makes planet earth unique or is there a deeper relation between the sizes, masses, distance ? Is there any mathematical equation involving the gravitational constant that describes this fact ?

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As Carlos explained the Moon's crescent is not about Earth's shadows. If you have a telescope take a look at Venus. When it is relatively close to Earth you will see a similar crescent. This is due to Venus being closer to the Sun than we are, and the crescent shows because only a small part of Venus' surface visible to us is sunlit. For example Mars never shows a thin crescent, because we are always mostly between Mars and Sun. OTOH, when viewed from the Earth, the Moon and the Sun do cover nearly the same angle. This is a coincidence. But it gives us awesome solar eclipses. –  Jyrki Lahtonen Dec 7 '13 at 14:03

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When you see a crescent moon, the dark part of the Moon is not the shadow of the earth: it is the “the dark side of the Moon”, i.e. the side of the Moon that is not currently lightened by the Sun.

Crescent moon

The inner curve of what you are actually looking is not a circle but an ellipse.

The shadow of Earth in the Moon is what you see in a Lunar eclipse, and it does not match the size of the Moon.

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