A first quarter moon shows half of its lighted hemisphere – half of its day side – to Earth.
But we officially call this moon a quarter and not a half because it is one quarter of the way around in its orbit of Earth, as measured from one new moon to the next.

Tonight's moon (March 30, 2012) from Muscat, Oman by EarthSky Facebook friend Priya Kumar. Beautiful Priya!
This moon appears half-lit to us, and half moon is a beloved name (although not an official one). Still, it’s good to recall that the illuminated portion of a first quarter moon truly is just a quarter. On the night of first quarter moon, we see half the moon’s day side, or a true quarter of the moon. Another lighted quarter of the moon shines just as brightly in the direction opposite Earth!
A first quarter moon rises at noon and is high overhead at sunset. It sets around midnight.
First quarter moon comes a week after new moon. Now, as seen from above, the moon in its orbit around Earth is at right angles to a line between the Earth and sun.
As the moon orbits Earth, it changes phase in an orderly way. Follow these links to understand the various phases of the moon.

Photo credit: Randy Miller
Understanding Moon Phases
Waxing Crescent
First Quarter
Waxing Gibbous
Full Moon
Waning Gibbous
Last Quarter
Waning Crescent
New Moon

good info,
awsome info i needed it for my projest tysvm.
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this helped so much thank you
thanks heaps, helped.
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thank-you for answers to my project
You’re welcome!
I’d like to discuse this paragraph (First quarter moon comes a week after new moon. Now, as seen from above, the moon in its orbit around Earth is at right angles to a line between the Earth and sun).
if moon in this picture (above) is set at right angle to a line between Sun and Earth, we have to observe the shadow of Earth on moon look like concavity. why we observe this shadow (on this picture) as a straight line?
you have to note that (at the time of Sun set) the moon is located obove the horizantal plane, but the Sun is under the horizantal plane. and the Moon’s orbit inclines about five degrees to Ecliptic plane.
it was great info for my phases of the moon project
suppp
To Mohammad T. Al-Tamimi: You asked “if moon in this picture (above) is set at right angle to a line between Sun and Earth, we have to observe the shadow of Earth on moon look like concavity. why we observe this shadow (on this picture) as a straight line?
You would only see the shadow of earth on the moon if earth were in the middle (causing an eclipse). When our Moon is in the middle it is like holding a ball between you and a lamp. Try it and see what it looks like. I hope this answered your question.
This was the best website used for our project! Thanks!
Why is this called a quarter moon? When it is full, it should be called a half moon if this is called a quarter moon. Just some words for thought. :)
Hi Bill, more food for thought …
Maybe it’s called a quarter moon because we’re only seeing a quarter of the entire globe of the moon.
Maybe it’s called a quarter moon because it’s one quarter of the way around in its orbit, as measured from one new moon to the next.
The origins of this name seem lost in history. I’ve never heard anyone answer this definitively.
All the best,
ES
thank…yahhh
love ya
[...] an explanation from the folks at EarthSky.org: [We] officially call this moon a quarter and not a half because it is one quarter of the way [...]
If a quarter moon is so called because it is a quarter of the way around the earth, then a full moon is only half way around and should be called a half moon. The naming of the phases is unnecessarily illogical. At a minimum, a full moon is the second quarter; the third quarter is really the first quarter on the flipside.
thank god this are the info i needed for my science project
[...] an explanation from the folks at EarthSky.org: [We] officially call this moon a quarter and not a half because it is one quarter of the way [...]
[...] phases of the moon. For our purposes today, we will only be considering the full moon, half moon (quarter moon), crescent moon, and new [...]