Donut (Torus) Calculator — Volume & Surface Area

सूत्र

How this is calculated

A torus is the mathematical name for a donut or ring shape — the solid formed by rotating a circle of radius r around an axis that is R units away from the circle's centre. Two inputs define it: R (the major radius, from the axis of revolution to the centre of the tube) and r (the minor radius, the radius of the tube itself). For a valid ring torus r must be strictly less than R so there is a hole in the centre.

The volume formula V = 2π²Rr² comes from Pappus's centroid theorem: the volume of a solid of revolution equals the cross-section area of the rotating shape (πr² for a circle) multiplied by the path length that the centroid travels (2πR), giving πr² × 2πR = 2π²Rr². The surface area A = 4π²Rr follows the same theorem applied to the perimeter of the circle (2πr) times the centroid path length (2πR).

Practical uses include calculating the capacity of toroidal tanks and O-rings, the volume of a ring-shaped garden bed, an inflatable swim ring, a bagel, or a donut-shaped architectural element. The outer diameter is 2(R+r), the inner hole diameter is 2(R−r), and the tube circumference is 2πr.