concentriccylinder

Concentriccylinder refers to a configuration of two or more cylinders that share the same central axis. The cylinders are coaxial, and their cross-sections perpendicular to the axis are circles centered at the same point. The radii and lengths may differ, but the axes coincide.

In the common two-cylinder case, the outer cylinder has radius R and the inner cylinder radius r with R > r. If both extend over a finite height h, the region between them forms a hollow, thick-walled cylinder. If the cylinders extend infinitely along the axis, the shape is an infinitely long coaxial tube.

Key geometric formulas describe a hollow cylinder of height h, outer radius R, and inner radius r. The volume is V = π h (R^2 − r^2). The curved (lateral) surface area is 2π h (R + r), representing the outer and inner cylindrical surfaces. If the end faces are included, the total surface area becomes S = 2π h (R + r) + 2π (R^2 − r^2) for a finite shell.

Applications and related concepts: concentric cylinders appear in engineering as thick-walled cylinders and pipes, where inner and outer radii define pressure, stress, and flow characteristics. They are a common model in fluid mechanics, for example in Couette flow between rotating coaxial cylinders, and in elasticity theory for thick-walled pressure vessels. In geometry and calculus, they provide straightforward examples for volumes and surface areas via cylindrical coordinates, washers, and shells.

See also: coaxial cylinders, annulus, thick-walled cylinder.