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Waxing Crescent Moon - As the Moon moves around the Earth, we get to see more and more of the illuminated half, and we say the Moon is waxing. At first we get a sliver of it, which grows as days go by. This phase is called the crescent moon.
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Quarter Moon - A week after the new moon, when the Moon has completed about a quarter of its turn around the Earth, we can see half of the illuminated part; that is, a quarter of the Moon. This is the first quarter phase. Also called 1st quarter Moon.
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Waxing Gibbous Moon - During the next week, we keep seeing more and more of the illuminated part of the Moon, and it is now called waxing gibbous (gibbous means "humped").
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Full Moon - Two weeks after the new moon, the moon is now halfway through its revolution, and now the illuminated half coincides with the one facing the Earth, so that we can see a full disk: we have a full moon. As mentioned above, at this time the Moon rises at the time the Sun sets, and it sets when the Sun rises. If the Moon happens to align exactly with the Earth and Sun, then we get a lunar eclipse.
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Waning Gibbous Moon - From now on, until it becomes new again, the illuminated part of the Moon that we can see decreases, and we say it's waning. The first week after full, it is called waning gibbous.
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Last Quarter Moon - Three weeks after new, we again can see half of the illuminated part. This is usually called last quarter. Also