Why is it shorter to travel a great circle route than a straight line route?

Why is it shorter to travel a great circle? As you will see, the great circle route is the closest you can get to a 'straight line' route on the globe. What looks to you like a 'straight line' route on a map really is not straight at all. Depending on how far apart two places are, the straight line between them will pass through some part of the Earth, and with the exception of relatively short tunnels will be completely impossible to travel. All flat maps are seriously distorted in some ways; you cannot project the surface of a globe onto a flat surface without compromising some detail. What looks like a straight route across your Mercator Projection will usually be longer than necessary, but see discussion. Countless great circle routes will look like straight lines, even on Mercator Projections. Putting the following ideas together will clarify what makes the great circle route the shortest possible surface or air route between two points on earth.

1. Every great circle cuts a globe, or a spheroidal planet like earth, into two equal halves, just as the equator does.
2. If you imagine any plane cutting right through and containing the center of the earth, then where the plane comes out to the surface will be a great circle.
3. The equator is a great circle, as are all of the lines of longitude (including the extensions of those lines on the other side of the poles, exactly 180 degrees of longitude over).
4. The number of possible great circles on the earth is limitless.

If you are standing on flat ground, on a circle you have drawn out that is about 5 feet in diameter, you can clearly see that your small portion of the circle is curved. The point under your left foot and the point under your right foot are connected by a curved arc of the circle. As you imagine the circle getting larger and larger, you can imagine the curve between your feet straightening out. The circle can't get any larger than a great circle, the circle that cuts the earth into two equal halves. This is why a great circle describes the shortest land route between two points on earth. This isn't noticeable in the distance between your feet, but with distances of hundreds or thousands of miles it becomes more and more important.

Any two points on earth's surface can be connected by the circumference of a great circle. This is easily demonstrated with a globe and a length of strong thread or a very thin cord. Connect any two cities with ends of the cord and you can take up slack until the cord is resting on the shortest surface path between them, the great circle.Check links.

If the two points are exactly opposite each other on the globe (they are called antipodes) like the north and south poles, than infinitely many great circle routes connect them.