Another question that the Flat Earthers cannot explain.
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There aren’t really any flat earthers. You are operating under a misconception. They are largely products and/or agents of a psyop.
The answer is, it doesn’t stay the same size. The sun does change (apparent) size during the day and over the course of the year. Wether the world is flat or spherical has no bearing on that. All things that recede appear to shrink (assuming no magnification/distortion in the way that obscures that optical law)
Has anyone observed and measured the sun changing size? How would that even work?
Yes.
It’s important to recognize that we are talking about apparent size change (though in the standard view of the sun being a star powered by fusion - the sun does change actual size as well)
The apparent size of the sun changes due to refraction/diffraction (typically at sunrise/sunset) as well as distance to us (typically over a yearly period, though obviously it is changing distance over the day too - so its apparent size is also changing daily even if we currently lack the precision to detect such a change)
Often it is done with an ND filter (to get rid of the glare and only measure the sun) and a photodetector (ccd/camera etc.).
Do you have any documentation?
Sure, but i would make you a less capable student - and risk further biasing you - if i spoonfed it to you.
Do you honestly doubt that the things which recede from us appear to shrink in size as a result? Do you honestly doubt that distortion caused by the air and things in it can (and does) affect the apparent size of the sun? Do you honestly believe that the sun is NOT changing distance from us (i.e. the observer) over the course of the day and/or year?
If the answer to all the above is, “No” - as i expect it ought to be ... then you don’t really need any “supporting documentation” in any case. Right?
Anyway, have you tried looking for such measurements already and failed to find them? Where have you looked, and what have you been looking for (search terms / etc.)?