About o Torus
What Is a Torus?
A torus is a three-dimensional geometric shape that resembles a donut or ring. It is formed by rotating a circle around an axis that is coplanar with the circle but does not intersect it. The torus is characterized by two radii: the major radius R (distance from the center of the tube to the center of the torus) and the minor radius r (radius of the tube itself).
Volume
The volume of a torus is calculated using the formula V = 2pi²Rr², where R is the major radius and r is the minor radius. This elegant formula comes from the Pappus centroid theorem, which states that the volume equals the cross-sectional area (pi r²) times the distance traveled by its centroid (2pi R).
Surface Area
The total surface area of a torus is SA = 4pi²Rr. This formula also derives from the Pappus theorem: the perimeter of the cross-section (2pi r) times the centroid path (2pi R). The outer surface (facing away from the center) is slightly larger than the inner surface (facing the center) due to the different radii of curvature.
Applications
Tori appear in many areas of science and engineering. In physics, tokamak fusion reactors use toroidal magnetic fields. In architecture, torus shapes appear in columns and moldings. In biology, torus-shaped molecules and structures exist. In everyday life, donuts, rings, and inner tubes are all torus shapes.