A circle.
However, a way of showing a sperical object on a flat surface, such as a map of the world, is called a projection. The most widely used example is the "Mercator Projection", the familiar classroom World map, although there are many others.
on a flat map
mercator
mercator
mercator
mercator
Its called a projection map
projection map
A system of mapping the round earth on a flat surface is called a map projection. Map projections are used to represent the Earth's curved surface on a two-dimensional map. Different map projections have different strengths and weaknesses depending on the purpose of the map.
A flat round object is a circle. Surface area of a circle = pi*radius squared.
A small flat copy of the Earth is called a map. It is a representation of the Earth's surface, showing geographical features and locations in a scaled-down form.
Its called a projection map
The flat surface that goes on and on in all directions has no boundaries. This flat surface is called infinite surface. Accordingly, the answer is that the flat surface that goes on and on in all directions is called an infinite surface.