Final answer:
The Egyptian tomb structure with a rectangular shape, sloping sides, and a flat roof is known as a mastaba. Mastabas preceded the construction of the famous step and smooth-sided pyramids and also mark the use of stone in their construction.
Step-by-step explanation:
The type of Egyptian tomb structure that is rectangular in shape, has sloping sides, and a flat roof is called a mastaba. Mastabas were constructed before the development of the more widely known step pyramids and smooth-sided pyramids. An architect named Imhotep, serving under Pharaoh Djoser, first used the mastaba as the basis for what would become the step pyramid, by stacking smaller mastabas atop an original stone mastaba. The resultant structure was a multitiered pyramid that marked a new era in Egyptian monumental architecture. A mastaba was typically built of mud bricks and later of stone, marking the first time stone was used in such buildings, which provided a durable alternative to mud brick construction. The inside of a mastaba could be elaborate, containing a long corridor with side chambers, decorated with bas reliefs and inscriptions resembling those found inside aristocratic homes, making it a 'house of eternity' for the deceased.