For very large structures, pyramids are easy. Structures the size of the Eygptian pyramids present a serious problem: their weight. The lower parts of the structure have to support the considerable weight of the upper parts. A pyramid-like shape, where the lower parts are wider, gives the lower parts the strength and stability to do so. Even loose granular materials can form stable pyramid-like shapes:
(in this case, a pile of sugar). Because the square blocks of a pyramid are more resistant to slipping than grains of sugar or sand, a pyramid can be built at a steeper angle than a loose sand pile or sugar pile. Still, if you make it too steep, it can collapse. This is believed to have happened the the Meidum pyramid:
during its conversion from a stepped pyramid to a larger smooth-sided pyramid
by the Old Kingdom (4th Dynasty) pharaoh Sneferu. Sneferu's more famous pyramid, the Bent Pyramid:
has an abrupt change in angle. This might have been a change of plan mid-construction in response to the collapse of the Meidum pyramid.
Given the engineering advantages of pyramid-like structures, it should be unsurprising to see pyramid-like large structures independently developed. If height is desired, then the slope of the sides should be as steep as possible. With small pyramids, which don't approach the limits of the forces that the stone can withstand, the sides can be quite steep, such as this example from the Sudan:
Note from the climber that this pyramid is much smaller than the large Egyptian pyramids.
Other than the slope, design choices include whether to make the sides smooth, like the classic Egyptian pyramid, or stepped:
There are many examples of stepped pyramids and pyramid-like structures, such as ziggurats in Mesopotamia, and stepped pyramids in the Americas (and the Egyptian example at Saqqara, above, and the original stepped version of the Meidum pyramid). The other choice is the shape of the base. While the usual choice is a square base, any generalised conical shape will work. There are some pyramids with circular bases in the Americas:
The stability of pyramidal structures is why the Great Pyramid was able to be the tallest man-made structure for many thousands of years. Lincoln Cathedral supposedly became taller in 1311, due to the addition of a tall spire on the main tower:
making its height 160m, more than the Great Pyramid's 146m. Note that the spire itself was pyramid-shaped - even for hollow structures, the wider base provides strength. The main spire collapsed in a storm in 1548 (the smaller spires of Lincoln Cathedral were still present in the 17th century but are now gone), while the more robustly-built Great Pyramid has survived many storms and earthquakes (although it has lost some height due to erosion, and is less than 140m tall now). The Great Pyramid didn't regain the title of Tallest Structure since St. Mary's Church in Stralsund, Germany had reached a height of 151m in 1495. The St. Mary's spire burned following a lighting strike in 1647, but the now-eroded Great Pyramid was a little shorter than Strasbourg Cathedral's 142m.
The Great Pyramid is an impressive structure, and a great achievement of logistics. From an engineering and architectural perspective, it is a far simpler structure than the cathedrals which later surpassed its height.
