Imagine standing (somewhere) on the Moon's surface. How big would the Earth look from the Moon, compared to the image of the Moon we see from Earth? What would be the ratio between the diameters of each body? How would this ratio be calculated? Any help with answering such a calculation would be highly appreciated.
There are a couple questions here. The ones you have asked are:
- What would be the ratio between the respective diameters?
- How would this ratio be calculated?
But I think you meant to ask:
- How much larger does the Earth appear when standing on the Moon than the Moon appears when standing on the Earth? (and the calculation)
For the answer to #1 (and #2), it is the ratio between the numbers (of the Earth's diameter of 12,742 km and the Moon's diameter of 3,474 km) which is $Earth:Moon = 3.668:1$ as @pela's comment mentions.
For the answer to #3, it is the same! The more interesting part is why. We know that the distance from the Earth to the Moon is the same as the Moon to the Earth, so it follows that when we are on one of them looking at the other we are gazing across the same distance. Next, we know that an object (approximately) appears larger in the same proportion as it is larger, so an object that is twice as wide or tall will appear twice as wide or tall. Since the Earth is 3.668 times as wide as the Moon, it will appear that much larger when viewed from the same distance, and we've already discussed why it is the same distance. So, that's it really. It is the ratio of the diameters.