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Q: What has a circular cross section when the cross section is taken parallel?

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The cross section of a sphere is most often a circle. When the cross section is taken parallel to the base of a cylinder, it is also a circle.

Only if the section is taken by a plane parallel to one of the faces.

A solid with a triangular cross section when the cross section is taken parallel to the base could be either a triangular prism, a triangular pyramid, or a triangular frustrum.. I've been sitting here trying to convince myself (without actually bearing down and trying to prove it mathematically) that any cross section of a regular tetrahedron (a special case of triangular pyramid) taken perpendicularly to the base is a triangle, and I THINK this is the case, but as I said I certainly haven't rigorously proven it; I'm just unable to come up with any obvious situation where this is not true.

The vertical cross section of a right vertical cone is a triangle if that cross section is taken from the vertex. Any other vertical cross section will reveal a hyperbola (with endpoints on the base of the cone). A link can be found below.

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The answer depends on the orientation of the pipe and the cut. Even if the cut is vertical, it can be along the axis (length) of the pipe, at right angles to it or at a slant. If the cut is along the axis, the cross section will be two rectangles where the length of the rectangle is the length of the pipe and the width is the thickness of the pipe. If the pipe has negligible thickness, this may be taken to be two parallel lines. If the cut is at right angles to the axis then the cross section will be an annulus which, when the thickness is negligible will become a circle. Finally, if the cut is skew, then you will get ellipses which will collapse to a single ellipse for negligible thickness.

pyramid

A cone or a pyramid.

A cone or a pyramid.

A pyramid or cone.

A tapered prism.

No. Some of the classic curves studied by mathematicians: ellipses, hyperbola are cross sections of a cone taken at an angle.

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