When we talks about the universe being flat we don’t mean flat like a sheet of paper rather than having volume like living inside a sphere. Think of the difference between a plane and the surface of a sphere. The curvature were talking about is an extension of that difference to three dimensions. It describes the geometry of that space.
In school you mostly learn flat geometry so the fact that there are others can be confusing at first.
Start with a square. Extend two opposite sides as lines towards infinity. If those lines get closer together as you look further away from your square your universe has positive curvature. If they get farther apart, it had negative curvature. If they stay the same distance forever, your universe has zero curvature (i.e., is flat).
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