halvcylindrical

Halvcylindrical is an adjective used to describe a shape, solid, or surface that is half of a right circular cylinder, obtained by cutting the cylinder with a plane that contains its axis. The term is often used in geometry, architecture, and engineering to specify a semi-cylindrical form.

In three dimensions, a right circular cylinder of radius r and height h oriented along the z-axis can be halved by the plane y = 0. The half-cylinder consists of all points with x^2 + y^2 ≤ r^2 and y ≥ 0, for z in [0, h]. Its boundary includes the curved semicylindrical surface x^2 + y^2 = r^2, y ≥ 0, z ∈ [0, h]; the two end semicircular faces at z = 0 and z = h; and the flat rectangular face along the cut, given by y = 0 with x ∈ [-r, r] and z ∈ [0, h].

Key measurements for a halvcylindrical solid include its volume and surface area. The volume is half the volume of a full cylinder, equal to (1/2) π r^2 h. The surface area comprises the curved semicylindrical surface (π r h), the cut-face rectangle (2 r h), and the two end semicircles (π r^2 in total), giving a total area of π r h + 2 r h + π r^2.

Halvcylindrical forms appear in various applications, such as architectural elements, troughs and ducts, or features in sculpture and industrial design. Variants include hollow half-cylinders and semi-cylindrical shells.