How far should we be from Earth to see a half Moon while it is seen as a crescent on the Earth?

When we see a crescent Moon that means it should be seen a half Moon from another position in the sky, because the terminator line (i.e. the line that separates the bright and dark areas of the Moon) is in fact a meridian. And a Meridian is a straight line which is seen curved when it's looked from an angle.

So when we see a regular narrow crescent Moon on the Earth, how far should we be from the planet in its orbit to be in front of that Meridian?

• Quick answer is 384,400 km, or the distance of the Moon to the Earth. You’d then be over the terminator and (except for horizon effect) you would see the moon exactly half-lit. Moving away from the Moon to see it whole and cancelling the horizon effect brings you further, but your question implied the minimum distance… Jun 28, 2023 at 0:33
• What about on the Earth's orbit? Jun 28, 2023 at 0:37
• Roughly 5.4 million km. Demonstration coming up. Jun 28, 2023 at 0:42
• The title asks about a full Moon, but the body of the question implies a quarter Moon. Which do you want? Jun 28, 2023 at 21:45
• I will edit the title. However, I don't recall I wrote Full Moon (also with a capital F!). Update: A user called eshaya has edited my post. Jun 28, 2023 at 22:17

• Can use Pythagorian Theorem: $d =\sqrt{(R^2 - (R-r)^2)}$. Where R is orbital radius of Earth and R is orbital radius of Moon. Jun 28, 2023 at 17:03
• @eshaya: Indeed. Your formula gives about 10 million kilometres. It seems that I measured wrong (forgot to multiply my drawing’s length by 2) yesterday; apologies. Snack Exchange: Yes, the formula is valid. The hypotenuse is the line from the Sun to the left part of my drawing, so $R$, while the long side of the triangle is from the Sun to the bottom right of my drawing, so $R - r$ indeed. Jun 28, 2023 at 22:29