We are hovering 250m over the equator of an alien planet which is homogenous and perfectly spherical, and a satellite with a perfectly circular orbit is traveling parallel to the equator. We know that the satellite takes 800 minutes to orbit the planet, and from our position, we can see 406 minutes of that orbit, or a little over half, before the satellite is eclipsed by the planet. While standing on the ground under the spot we were hovering, we can see the satellite for exactly half of the orbit, or 400 minutes. What is the approximate diameter of the planet? Explain why the answer cannot be exact.
I don't wan't an answer, just insight into how to solve this problem, and which mathematical tools I should be looking to use. My current idea is to use the perspective shift as a form of measurement, roughly constructing the circumference of the planet into an n-sided polygon, and then summing the sides and dividing by pi. I don't exactly know how to get from the 250 ft elevation to the length of each side of this polygon, but I feel as though there'd be a way. Not sure if this intuition is correct though, so I'd be interested what those more experienced with mathematics would think. This is my first post on a forum like this, so excuse me if it's ill-placed.