For example, a tape measure, normally coiled for storage, is used to measure distance along a straight path or around a curved path. As an example of that, one can determine the circumference of a tree by wrapping the tape around the tree and reading at the appropriate place. Equivalently, the measure can be taken by wrapping a rope around the tree, and then measuring it afterwards when it is laid out straight. That is, circles map easily to line segments. And arc length on a circle is directly analogous to length of a line segment.
Spheres are round and planes are flat.
As we associate length with line segments, we associate area with portions of a plane. And we associate concepts of line and angle with both spheres and planes. But using a flat surface to represent a sphere, or portion of it, involves risks not present in the circle-line relationship. The desire to view the surface of Earth on a planar surface probably arises from both historical and pragmatic reasons. Regardless of the shape of an object, illustration of it has been on planar media through many stages, from drawing on sand and walls of caves to drawing and writing on papyrus and animal skins, to printing on paper, to display on electronic screens that are essentially flat.