A pyramid or square hip roof is shaped like a pyramid with four equal-sized triangles rising to meet in the center. On a rectangular house, the roof will have two rectangular sides and two trapezoidal sides, all rising at the same pitch, typically at 45 degrees.
This roof style is named after the French architect Francois Mansart. The mansard roof has two different roof angles, with one steeper than the other. The upper roof slope is usually not visible from the ground. The degree of pitch on the mansard roof can be as little as 20 degrees and as much as 70 degrees. The advantage to this style of roof is the added internal space, which can be converted into living space.
A Dutch gable is a a combination of a hip roof and a gable roof with a pitch that can run from 15 degrees to 30 degrees. This is constructed as a hip roof with a full or partial gable set at the end of a ridge. This roof style has the benefits of both a gable and a hip roof.
The half-hip style of roof is often seen in northern European architecture. It is a gabled roof with a small hip roof at the top. A gablet roof is the reverse of the half hip. It is a hip roof with a gabled roof above. This roof, like the dutch gable, can run anywhere from 15 to 35 degrees.