I really don't understand what a plane is. What would be an example of it in real life?

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In math, a plane is a flat surface that extends forever in all directions. In real life, you're unlikely to encounter such an infinite surface, but the top of a table or the surface of an ice skating rink are examples of (limited) planes. If you've ever been in the middle of Kansas and looked around you at the landscape, the flat "plains" give the appearance of a plane (although in reality, of course, the earth's surface is curved).