It looks like half the moon is illuminated, not 25%, and to our eyes, half of the moon's surface facing us is illuminated. How much of the moon's entire surface is illuminated?
Regardless of what we see in the sky, exactly one half (50%) of the moon's surface is illuminated at all times. What we see depends on the positioning of the moon relative to the earth and the sun. Consider the full moon below.
To our eyes, 100% of the side of the moon facing us is illuminated. Yet only 50% of the moon's entire surface is illuminated. The moon is positioned relative the the earth and the sun in such a way that we see all of that 50%.
So why a quarter moon? Nowhere in the discussion above does 25% come into play! A quarter moon has nothing to do with seeing 25% of the moon's surface illuminated or the moon's surface being 25% illuminated. The moon's surface is ALWAYS 50% illuminated. The phrase 'quarter moon' has to do with where in the lunar cycle the moon is. The start of a lunar cycle occurs when the moon is in its new phase. Next comes waxing crescent, first quarter, waxing gibbous, full, waning gibbous, third quarter, and finally waning crescent before heading back to a new moon. A first quarter is one quarter of the way through this cycle and a third quarter is three quartesr of the way through the cycle. This is where the quarter phases come from, not the % observed to be illuminated or the % of actual illumination.