Calculations for a catenoid, or the solid of revolution of a catenary curve around the x-axis. This contrasts with the catenary dome, where the catenary curve rotates around the y-axis. This shape is formed from the underlying catenary curve of the form y = a*cosh(x/a) in the interval x ∈ [ -b ; b ], which rotates around the x-axis. Since the curve does not intersect the axis of rotation, a rotationally symmetric hollow body with two circular boundary surfaces is created.
Enter the shape parameter or minimal radius a (a>0) and the maximum input value b. Choose the number of decimal places, then click Calculate.
Formulas:
pi:
Shape parameter, minimal radius a and height have the same unit (e.g. meter), lateral and surface area have this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit -1.
The catenoid resembles the one-sheeted hyperboloid but is based on different mathematical functions. The hyperboloid is described by quadratic equations, while the catenoid arises from the catenary curve and therefore contains hyperbolic functions. Unlike the hyperboloid, the catenoid is a minimal surface. It is the surface with the smallest possible area between two boundary circles, provided the surface can stretch freely between these boundaries and no additional restrictions are imposed. A cylinder with a smaller lateral surface does not fulfill this property because its surface is flat in the longitudinal direction and therefore does not possess the balanced curvature required for a minimal surface.
Therefore, the catenoid is used in membrane construction, where the material must be under uniform tension to prevent wrinkles or tears. Due to these special geometric properties, the catenoid also plays a role in differential geometry and physics, for example, in the study of minimal surfaces and equilibrium forms of stretched membranes. For instance, if you immerse two wire rings in soapy water and pull them apart, the soap film forms a catenoid.