No it isn't because it doesn't have any points.
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A cylinder has a circular cross-section whereas a cuboid has a quadrilateral cross-section.
A quadrilateral rectangle or a square
No. A quadrilateral is any four-sided figure. A trapezoid is one example of a quadrilateral.
Yes, such a quadrilateral can exist.
A quadrilateral is a polygon with four sides. A parallelogram is a quadrilateral in which the opposite sides are parallel.