Egypt (3000 BC - 500 BC)
The geometry of Egypt was mostly experimentally derived rules used by the engineers of those civilizations. They developed these rules to estimate and divide land areas, and estimate volumes of objects. Some of this was to estimate taxes for landowners. They also used these rules for construction of buildings, most notably the pyramids. They had methods (using ropes to measure lengths) to compute areas and volumes for various types of objects, various triangles, quadrilaterals, circles, and truncated pyramids. Some of their rule-based methods were correct, but others gave approximations. However, there is no evidence that the Egyptians logically deduced geometric facts and methods from basic principles. And there is no evidence that they knew a form of the "Pythagorean Theorem", though it is likely that they had some methods for constructing right angles. Nevertheless, they inspired early Greek geometers like Thales and Pythagorus. Perhaps they knew more than has been recorded, since most ancient Eygptian knowledge and documents have been lost. The only surviving documents are the Rhind and Moscow papyri.
Ahmes (1680-1620 BC)
wrote the Rhind Papyrus (aka the “Ahmes Papyrus”). In it, he claims to be the scribe and annotator of an earlier document from about 1850 BC. It contains rules for division, and has 87 problems including the solution of equations, progressions, areas of geometric regions, volumes of granaries, etc.
Anon (1750 BC)
The scribe who wrote the Moscow Papyrus did not record his name. This papyrus has 25 problems with solutions, some of which are geometric. One, problem 14, describes how to calculate the volume of a truncated pyramid (a frustrum), using a numerical method equivalent to the modern formula: , where a and b are the sides of the base and top squares, and h is the height.
The book Mathematics in the Time of the Pharaohs gives a more detailed analysis of Egyptian mathematics.