On three-dimensional geometric objects with round or curved faces
In this context, round shapes are defined as those exhibiting at least some curvature, even if other parts are straight, analogous to the two-dimensional round shapes. The most familiar and regular spacial round shape, essentially the epitome of roundness in three dimensions, is the sphere. Cutting a sphere in half yields a hemisphere, which has one curved surface and one flat surface, but no straight edges. Other well-known shapes of this type include the cylinder and the cone. The latter serves as an example of a round shape that also features a pointed tip. Examples of round shapes with straight edges include the cylindrical sector, where calculations are relatively straightforward, and the cylindrical wedge, which involves significantly more complex formulas.
The curvature of round spatial objects can and often does derive from a circle, though this is not strictly required. Counterexamples include the spheroid and the ellipsoid, which are based on ellipses, as well as the paraboloid and the hyperboloid, which are based on other conic sections. Of course, the more complicated the underlying curves defining the shape, the more complicated the calculations become. These often rely on the integration of the curves in question. The spindle serves as an example of a shape requiring such integration.
A generalization covering many three-dimensional round shapes is the solid of revolution. This encompasses shapes that possess an axis of rotation, whose cross-section perpendicular to that axis is always a circle or an annulus. The universal formulas for solids of revolution require the centroid of a curve, of a surface area and a curve length, values that can be very difficult to determine, depending on the shape's complexity.