On average, how far away is the Moon from Earth, measured in kilometres?

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Answer

About 380,000 km

The Moon drifts away from us by about 3.8 cm every year, roughly the speed at which fingernails grow.

How to estimate it

The Moon looks about half a degree wide in the sky, and an object that small sits roughly 115 times its own width away (57 divided by 0.5). The Moon is about a quarter of Earth's diameter: 12,800 / 4 is roughly 3,500 km. So the distance is about 3,500 × 115 = 400,000 km. The measured average is about 380,000 km.

Source: NASA

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