30 degrees south of the equator on the east coast of australia is the small town of red rock. If we take that point of 30 degrees south and 153 degrees and 13 minutes east. Then, if we take a point on the west coast of australia, it has the coordinates of 30 degrees south and 115 degrees east. The difference in longitude is 38 degrees and 13 minutes.
If you were to travel directly from the point in Red Rock and travel directly west until you reached our second point, you would cover a distance over land of 3,687km.
So what happens if you travel along the same 30 degrees south latitude line for 38 degrees and 13 minutes on the flat earth map?
The 30 degree south line is 13,322km away from the geographic north pole and thus if distances are the same, it is the same on the flat earth map. We know the longitudinal and thus angular separation between our two points at this latitude: 38 degrees and 13 minutes, or 38.217 degrees.
The length of an arc on a circle subtended by an angle, alpha, is:
L = a π r /180
When we put in our numbers:
L = 38.217 π (13,322)
180
L = 1,599,469/180
L = 8,886km
We find that the two points on the flat earth surface 30 degrees south of the equator and separated by 38 degrees and 13 minutes are 8886km apart. This is 2.41 times greater than the real distance between the two points.
But you say that distances are the SAME as in real life. In this case, 3,687km. In that case, you must say that separation in longitude on the flat earth must be considerably smaller than in real life. Thats how geometry works. If we make L 2.41 times smaller, we must make alpha 2.41 times smaller which means on the flat earth the difference in longitude between our points would be 15.85 degrees .
But it isn't.
The distances cannot be the same. Its basic geometry.