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I was reading the article Moon Phases on HowStuffWorks. In the picture, each moon has a dark green area which represents the shadow of the Sun. How is this shadow formed and why is this important?

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The half of the Moon not represented by the dark green semicircle is directly illuminated by the Sun. The dark green half is away from the Sun, in solar "night." [Note that this does not mean that it is actually dark there, because the Earth casts a lot of light onto the surface of the Moon as well.] The light and dark areas of Sun illumination determine the phase of the Moon as seen from Earth.

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That is a very odd diagram - the area of the moon shown as dark is not necessarily dark: it just represents the side we cannot see.

The green area is meant to represent the side unlit by the sun, leaving the lit section as that which we can see from Earth and lit by the sun.

It is obviously done this way to help you understand why the phases look like they do, but using green for shadow, and dark for not visible is confusing.

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