prolate

Prolate is an adjective used in geometry, physics, and related disciplines to describe a body that is elongated along its principal axis; its opposite is oblate, which is flattened at the poles. The term is commonly applied to ellipsoids and spheroids.

In geometry, a prolate spheroid is a special case of an ellipsoid obtained by rotating an ellipse about its major axis. If the semi-major axis is a and the two equal semi-minor axes are b, with a > b, the figure is prolate. The volume is V = 4/3 π a b^2. The surface area has a closed form in terms of the eccentricity e = sqrt(1 − (b^2/a^2)) and is S = 2π b^2 [1 + (a/(e b)) arcsin e]. As b approaches a, the prolate spheroid approaches a sphere.

In physics and astronomy, the term prolate describes elongated shapes or deformations. In nuclear physics, nuclei may exhibit prolate shapes in certain excited states, characterized by a positive quadrupole deformation parameter β2. In astrophysics and planetary science, galaxies, dark matter halos, and some asteroids or stars can be modeled as prolate ellipsoids when their mass or density distributions are stretched along one axis.

Related terms include oblate, which describes flattening at the poles; ellipsoid and spheroid, which are broader categories of three-axis shapes. A common everyday analogy is the rugby ball or American football shape, which resembles a prolate spheroid when cross-sections are circular in the minor axes.