Every map is a lie
You can't flatten a sphere without breaking something. The question is what you choose to break.
I got into an argument last week about the size of Africa. Someone was convinced it was "about the same" as Greenland. They'd seen it on a map. The map was wrong, of course, but try telling someone the map they've looked at their whole life has been lying to them. They look at you like you just said the sky is plaid.
But it has. Every flat map is a distortion. Not roughly, not around the edges. The geometry simply doesn't permit it. You cannot take the surface of a sphere and lay it flat without warping something. Gauss proved this in 1827. Mapmakers have been coping ever since.
Peel an orange
Seriously, go do it. Peel an orange and try to press the skin flat on a table. It cracks. It bunches. It refuses. A sphere's surface has intrinsic curvature that a flat plane doesn't, and no amount of cleverness gets around that.
So every projection picks its sacrifice. You get to keep shapes, or areas, or distances, or directions. Pick one, maybe two. The rest get mangled.
Mercator
The one in your head. Keeps shapes and angles intact, which is why sailors used it. The cost is that anything near the poles gets blown up to absurd proportions. Greenland becomes a continent.
Gall-Peters
Gets the areas right. Africa is finally the right size. But everything looks like it's melting. It's ugly, honestly, but it's honest.
Robinson
The compromise pick. Nothing is accurate, nothing is terrible. National Geographic used it for a while, which is about the most lukewarm endorsement a projection can get.
Winkel Tripel
Current National Geographic standard. Wrong about everything, but less wrong than anything else. I have a soft spot for it.
How bad is it, really
Bad. On a Mercator map, Greenland looks roughly the same size as Africa. In reality you could fit fourteen Greenlands inside Africa. Fourteen. That's not a rounding error, that's a completely different planet.
Actual land area (million km²)
Hit that toggle. Watch Greenland balloon to 73% of Africa's bar. That's what's sitting in your head from every classroom wall and Google Maps session you've ever had.
The Peters affair
The Mercator inflates Europe and North America. It shrinks Africa and South America. For centuries, the standard classroom map literally made the global north look bigger than it is. You can draw your own conclusions about that.
Arno Peters did, loudly, in 1973. He promoted his "new" equal-area projection as a corrective to colonial cartography. Cartographers hated him. Partly because his projection makes continents look like wet laundry on a line, partly because he wasn't saying anything they didn't already know, and partly because James Gall had invented basically the same projection in 1855. Peters just had better PR.
The cartography establishment fought back with the energy of people defending their doctoral theses, which many of them literally were. The American Cartographic Association eventually passed a resolution against all rectangular projections. An entire professional body officially said "stop using rectangles." I think about this a lot.
Globes
The honest answer is just use a globe. Everyone hates hearing this.
If you need something flat, the Winkel Tripel is probably the least bad option. It lies about everything a little bit instead of lying about one thing a lot, which is somehow the most truthful approach. National Geographic switched to it in 1998. I'd hang one on a wall.
That person I argued with pulled up a globe app on their phone, zoomed to Africa, and went quiet for about ten seconds. Then: "Oh."
Yeah. Oh.