heighttoradius

Height-to-radius, often written as h/r, is a dimensionless geometric parameter that expresses the ratio between an object’s height and its base radius. It is most commonly applied to axisymmetric bodies with circular bases, such as cylinders, cones, and domes, where height is the perpendicular distance from the base to the top and the radius is the radius of the base circle. The ratio is calculated simply as h divided by r.

In practice, height-to-radius is used to compare proportions independent of size. For a cylinder or cone, h/r provides a quick sense of how tall or slender the object is. The same ratio can be applied across disciplines, including architecture, engineering, product design, and computer graphics, to assess aesthetics, space, and functional behavior.

The ratio has practical implications for performance and stability. A higher h/r indicates a taller, more slender shape, which can influence wind loads, stiffness, and buckling tendencies in structural applications. It also affects volume and surface-area characteristics: for a cylinder, volume V = π r^2 h and surface area S = 2 π r (r + h). Modifying h while keeping r constant changes these properties in predictable ways, governed by basic geometry.

Examples include a cylinder of height 20 units and radius 5 units (h/r = 4) and a tall tower of height 100 units with radius 2 units (h/r = 50).

Limitations: the h/r ratio alone does not capture curvature, taper, material properties, end conditions, or aerodynamic factors. Context is essential for engineering and design decisions.

See also: aspect ratio, slenderness, radius of curvature.