Torus (Doughnut) Volume and Surface Area
The volume and surface of a torus - an O-ring, an inner tube or toroidal float, a doughnut tank, or a coil of tubing.
Example
You enter
- Ring centerline diameter Dc (in) 12
- Tube diameter dt (in) 2
You get
- Volume 118.435
- Surface area 236.87 in^2
Details, formula, and sources
Pappus theorem: V = 2 pi^2 R r^2, surface = 4 pi^2 R r, with R = Dc/2 (ring centerline radius) and r = dt/2 (tube radius). A 12 in ring of 2 in tube holds 118 in^3 (about half a gallon) with 237 in^2 of surface; a 36 in ring of 6 in tube jumps to 3,198 in^3 (13.8 gal). The tube must be no fatter than the ring (dt <= Dc) or the doughnut closes its hole. Volume in in^3, ft^3, and gallons. A partial fill, a non-circular tube, and wall thickness are separate. A takeoff aid; verify against the drawing.
R = Dc/2, r = dt/2; V = 2 pi^2 R r^2; surface area = 4 pi^2 R r. Requires dt <= Dc.
The torus volume V = 2 pi^2 R r^2 and surface area 4 pi^2 R r (Pappus's theorem; standard solid geometry as in Machinery's Handbook, Industrial Press), by name; public domain.
Pure solid geometry, public; the centerline and tube diameters are user-supplied measurements.
Estimate. AHJ and licensed professional govern.
Field names used by the API: center_diameter_in, tube_diameter_in, volume_in3, surface_area_in2
- Pappus V = 2 pi^2 R r^2, SA = 4 pi^2 R r (cross-section swept around the ring path)Pappus's theorem
- Ring torus tube no fatter than the ring (dt <= Dc); otherwise horn/spindle torusgeometry
- Scope circular tube, full torus; partial fill, non-circular tube, wall thickness are separatescope of this tile