A compass is used to construct a perfect circle of 360 degrees.
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Euclid came up with the most common form of geometry.
The ancient Egyptians displayed their considerable grasp of geometry in a number of ways. What is arguably the most obvious is the way geometry was used in the construction of the pyramids and other buildings and monuments. Certainly they laid out their cities using geometric principles. It was Ahmes (actually Ahmose), who left us with a fine approximation of the value of pi from that time. There are other examples, but none with the drama and "recognition value" of the geometric applications in religious architecture and construction first cited. Wikipedia has a bit more posted, and a link to their article can be found below. In other words they used geometry by building pyramids.
Aristotle considered geometry one of the most important sciences, and did some work with point and line planar geometry. He also used geometry as a way into sciences where he did more work, like optics and mechanics.
Yes. When constructing a garment geometry is one of the most important part of the design.
Most of the time, a tool called a "compass" is used to draw perfect circles. This compass is not to be mistaken with a directional compass.