domeandcylinder

Dome and cylinder refers to a composite solid formed by a right circular cylinder capped on top by a hemispherical dome. This combination is used in design and engineering contexts where a tall, rounded upper profile is desirable, such as in architectural elements, storage tanks, or decorative structures. The two parts are typically joined along a common circular cross-section.

In the common configuration, both parts share the same base radius r. The cylinder has height h, contributing to the overall height of the object, which is h plus the radius of the dome (for a hemisphere, the dome height is r). The result is a closed solid with a curved top and a flat bottom, though variants exist where the bottom is not exposed.

Volume is the sum of the cylinder’s volume and the dome’s volume. For a hemispherical dome of radius r, V = π r^2 h + (2/3) π r^3. If the dome is a general spherical cap of height h_cap, the cap volume is (1/3) π h_cap^2 (3r − h_cap), giving V = π r^2 h + (1/3) π h_cap^2 (3r − h_cap).

Surface area depends on whether the bottom is exposed. If the bottom base is included (closed solid), the surface area is A = 2 π r h (cylinder) + 3 π r^2 (hemisphere curved surface plus bottom base). If the shape rests on the ground with an hidden bottom, A = 2 π r h + 2 π r^2, since the base and the circular interface are not exposed.

See also: Dome, Cylinder, Spherical cap, Hemisphere.