Question

In what map projection is the actual shape of regions and countries preserved?

Answer 1

The spherical surface of the Globe (ellipsoid, globe) can not be expanded on the map plane without distortion.

Inevitably, there are deformations-compressions and stretches, different in size and direction. This is why there is a non-constant scale of lengths and areas on the map. All map projections have distortions.

Inevitably, there are deformations-compressions and stretches, different in size and direction. This is why there is a non-constant scale of lengths and areas on the map. All map projections are distorted. World maps are usually made in cylindrical, pseudo-cylindrical, and polyconic projections. Secant cylinders are often used to reduce distortion, and pseudo-cylindrical projections are sometimes given with breaks on the oceans.

Ideally, it's a globe.