All roofs, even ones that look flat, need to slope to some degree so that snowmelt and rainfall can drain off. But beyond that basic requirement, architects and builders have a lot of leeway, and they've used that creative freedom to invent a wonderful array of roof designs.
Octagon roofs have eight sides, as the term implies. Builders often choose this shape when they need to roof a building that's round or nearly round. Instead of dealing with the awkward fractions that another shape might require, they simply divide 360 degrees into half, then half again, then half a final time, winding up with the 22 1/2-degree angle that's needed to build an eight-sided roof.
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