Why is a great circle the shortest distance between two distant places on earth?

Answer 1

Yes, if you are talking about two points on earth's surface. The great circle can be thought of as roughly similar to a circle of longitude, or to the equator. It is the largest circle on the globe that can be drawn containing the two points in question. Why is this important? Consider the fact that the larger a circle becomes, the closer a section of the circle resembles a straight line. If you imagine a circle that is infinitely large, you would not be able to distinguish a section of it from a straight line drawn between the end-points. So when you have drawn the largest circle you can that contains two points on earth, you have come as close as you can to approximating a straight line between them (without digging). To people who are not familiar with this idea, seeing a 'great circle route' drawn out on a Mercator projection seems impossible. Map projections have to sacrifice some important detail, because you cannot map a three-dimensional globe onto a two dimensional surface.

Answer 2

A circle is not the shortest distance between two distant places on earth, but a pseudo-arc (a curve, or a piece of an oval) is, generally, the shortest distance between two distant points if one has no means of burrowing through the earth (or using a submarine if the distant points are on opposite shores of an ocean).

The reason is that the earth is a sphereoid, like a ball (but slightly fatter around the equator, and rough with mountains and oceanic trenches). Without piercing through the earth itself, someone trying to travel between two points on the surface of the earth must travel along the surface of the earth.

Imagine you have a ball, and two points on opposite sides of the ball ... if you were to draw a line between the points along the ball's surface, the 'line' would be curved since the surface of the ball is curved; the only true straight line would be a hole bored through the ball that pierced both points. Piercing the Earth is, of course, impractical and thus we are, more or less, restricted to traveling pretty much along the earth's surface which will always be an arc or curve (with bumps to surmount or circumvent topographical obstacles).