How as an individual can I know if the Earth is a sphere or a flat disc? What experiment can I do that doesn't involve trusting information from a 3rd party that would prove what the geometry really is?

Then, if you really want to know about this, be an astronaut and go outside our planet and check if the world is a sphere or a flat disc on your own. I think there is no astronaut here in the forum. LOL. But if there is someone who really knows this then maybe he/she can answer you here, I am just suggesting what you will need to do. LOL.
