The Egyptians didn’t just start building perfectly sloped pyramids. There are quite a few early pyramids that are interim phases, and some that, for various reason…didn’t work. They began with the mud brick Mastaba tomb, which is flat-topped with sloped sides, and there’s a lot of them.
The early pyramids (3rd Dynasty) are stepped pyramids, that is they have sides with very little slope, but with several layers that are significantly smaller as they go up, like stacked boxes, or rather several Mastabas stacked one upon the other. The Egyptians may have gotten the idea from Mesopotamian Ziggurats, as well. Over time, they toyed with how much to slope the side walls could take. The best example is Djoser’s Pyramid.
Then they tried some smaller smooth “provincial” pyramids with varying slopes. Probably built in the 3rd or 4th Dynasty, maybe working up to larger pyramids.
The first smooth-sided large pyramid (4th Dynasty) would have been the Sneferu or Meidum stepped Pyramid, which apparently was intended to be constructed first as a stepped pyramid and then was to be clad in a sloped casing to make it smooth sided. It partially collapsed during construction.
Then there’s the bent pyramid of Sneferu, which likely was intended to be a single smooth slope, but became unstable at the original angle, so there’s a noticeable ‘bend’ in the slope where the top bits are much more sloped, and so weighted less.
So, Senferu was intent on building a proper smooth-sided giant pyramid, and finally managed it with the Red Pyramid of Snefrou. It’s not as steep-sided as later pyramids, but at least it stayed up. Then his son, Khufu, managed to build a proper pyramid, and the 4th Dynasty built most of the well-preserved pyramids. Even then, they had a failure or two.
Then the Middle Kingdom tried mud bricks again, and while they might have been completed, didn’t last. Most barely resemble a pyramid.
So how did they know how to build them? Trial and error.
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