Surface of revolution in the context of "Cylinder (geometry)"

An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's shape and size, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different reference ellipsoids have been used as approximations.

It is an oblate spheroid (an ellipsoid of revolution) whose minor axis (polar diameter), connecting the geographical poles, is approximately aligned with the Earth's axis of rotation. The ellipsoid is also defined by the major axis (equatorial axis); the difference between the two axes is slightly more than 21 km or 0.335%.

A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circular symmetry.

If the ellipse is rotated about its major axis, the result is a prolate spheroid, elongated like a rugby ball. The American football is similar but has a pointier end than a spheroid could. If the ellipse is rotated about its minor axis, the result is an oblate spheroid, flattened like a lentil or a plain M&M. If the generating ellipse is a circle, the result is a sphere.

In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary). The surface created by this revolution and which bounds the solid is the surface of revolution.

Assuming that the curve does not cross the axis, the solid's volume is equal to the length of the circle described by the figure's centroid multiplied by the figure's area (Pappus's second centroid theorem).

Axial symmetry is symmetry around an axis or line (geometry). An object is said to be axially symmetric if its appearance is unchanged if transformed around an axis. The main types of axial symmetry are reflection symmetry and rotational symmetry (including circular symmetry for plane figures and cylindrical symmetry for surfaces of revolution). For example, a baseball bat (without trademark or other design), or a plain white tea saucer, looks the same if it is rotated by any angle about the line passing lengthwise through its center, so it is axially symmetric.

Axial symmetry can also be discrete with a fixed angle of rotation, 360°/n for n-fold symmetry.

In geometry, a torus (pl.: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut.

If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution, also known as a ring torus. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a spindle torus (or self-crossing torus or self-intersecting torus). If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is called a toroid, as in a square toroid.

In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution.

The theorems are attributed to Pappus of Alexandria and Paul Guldin. Pappus's statement of this theorem appears in print for the first time in 1659, but it was known before, by Kepler in 1615 and by Guldin in 1640.