You must be familiar with the idea that the shortest distance between two points is a straight line. It is one of the fundamentals of Euclidean geometry and can be proved using the calculus of variation or the principle of least action. But there is an important assumption: the surface is flat.

Let us assume a sphere (it has a non-flat surface). If the sphere is large enough, it can provide a flat surface at tiny distances. Keeping this in mind, we mark two points on the opposite sides of the sphere. What is the shortest possible distance between those two points? If your answer is “the same as the diameter of the sphere.”, that is right. Now, let me reiterate my question. What is the shortest possible distance between those two points if you were restricted to move along the surface of the sphere?  The curved line joining the two points would have a length of ΠR and is called a geodesic. 

The idea of a curved line being the shortest distance between two points is not one from Euclidean geometry. Hence, some rules of Euclidean geometry are no longer valid.

Consider now the triangle on the surface of the Earth made up of the Equator, the line of 0 degrees longitude, the line of 90 degrees longitude east. Both of the longitude lines make an angle of 90 degrees with the Equator. The two lines also meet each other at the North pole at an angle of 90 degrees. The angles of this triangle add up to 270 degrees. Hence, the statement ‘Euclidean geometry is only good for flat surfaces’ is right because 270 is not equal to 180.