Torus Volume & Surface Area Calculator
Compute the volume and surface area of a torus (the donut or ring shape) from its two radii: the major radius R (centre of the hole to centre of the tube) and the minor radius r (tube radius).
units
units
V = 2π²Rr²
V =
394.784 units³- 1
π²
π × π = 9.869604 - 2
Square the tube radius
r² = 2² = 4 - 3
Volume V = 2π²Rr²
2 × 9.869604 × 5 × 4 = 394.7842By Pappus’s theorem: circle area πr² times centroid path 2πR.
How does this calculator work?
A torus (donut) with major radius R and tube radius r (r < R) has volume V = 2π²Rr² and surface area A = 4π²Rr. Both formulas follow from Pappus's theorem: multiply the circle's area/perimeter by the path its centre travels around the axis (2πR). Enter R and r in any consistent length unit.
Formula
How this is calculated
A torus is the three-dimensional surface generated by rotating a circle of radius r around an axis in the same plane at distance R from the circle's centre. The two critical measurements are the major radius R — the distance from the centre of the torus to the centre of the circular tube — and the minor radius r, which is the radius of the tube itself. The condition r < R ensures the tube does not intersect or cross the central axis.
The volume formula V = 2π²Rr² follows from Pappus's centroid theorem: the volume swept by a plane figure equals the area of the figure (πr² for a circle) times the distance travelled by its centroid (2πR). The surface area A = 4π²Rr is derived the same way: the circumference of the circle (2πr) times the centroid's path (2πR).
The outer diameter of the torus is 2(R + r) and the inner hole diameter is 2(R − r). These formulas apply to a ring torus (r < R); a horn torus (r = R) touches the axis, and a spindle torus (r > R) self-intersects — both are excluded by the r < R guard. Units are consistent throughout: any length unit gives volume in that unit cubed and area in that unit squared.
Frequently asked questions
The major radius R is the distance from the centre of the whole torus to the centre of the tube (the "ring size"). The minor radius r is the radius of the tube cross-section (the "tube thickness"). Think of R as how big the donut is overall and r as how fat the dough is.
When r ≥ R the tube reaches or crosses the central axis of rotation, creating a self-intersecting or degenerate shape (horn or spindle torus). The volume formula V = 2π²Rr² still gives a number, but it no longer represents a simple hollow ring, so the calculator restricts to the ring torus case.
Common examples include a donut, a bagel, an O-ring seal, a life buoy, a rubber band, a tire inner tube, a coil of rope, and in physics, a toroidal solenoid or fusion plasma chamber (tokamak). The torus also appears in mathematics and topology.
Also known as
TG we-Calculate Editorial Team. (2026). Torus Volume & Surface Area Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/torus-volume-calculator
TG we-Calculate Editorial Team. "Torus Volume & Surface Area Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/torus-volume-calculator.
TG we-Calculate Editorial Team, "Torus Volume & Surface Area Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/torus-volume-calculator
@misc{wecalculate_torus_volume_calculator, title = {Torus Volume & Surface Area Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/torus-volume-calculator}}, year = {2026}, note = {TG we-Calculate} }
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