The polar projection does not distort measurements that are directly north/south. For that map, the distances along longitude lines match those on a globe. In other words, the distance from the north shore of Australia to south shore matches on both globe and north pole projection, the UN/most common FE map. So you can determine the north/south scale using 8000 mile diameter FE map. Distance from equator to north pole is the same on globe and FE polar projection map.
If you use that scale to measure the width of Australia, globe works, FE polar projection has Australia too wide, wider than USA. GPS, astral navigation, odometer, airline schedule, geometric calculation of distance on a sphere, all match RE and do not match the scaled measurement on FE polar projection map. Either FE does not match reality, or measurement is broken.
Perhaps this is why published FE maps never have scales. It has been suggested that FE rulers need to be bendable and stretchy. This flies in the face of the idea of measurement, which scientists go to great lengths to make constant and measure accurately. Measurement is not bendable or stretchy, that violates the very definition and concept of it.
If you want to portray the earth as a different shape than it actually is, you will inevitably need bendable and stretchy rulers. So here's the deal: RE - rulers can be straight and constant and distances match observed reality. FE - distances do not match, so measurement must be distorted to match observed reality.