Torus Volume
Calculator

A torus (a donut shape) needs two radii: the tube radius (r), the radius of the circular cross-section, and the central radius (R), the distance from the torus's center to the middle of the tube. Both must be entered correctly and in the same units, or the result will be wrong.

Volume is V = 2π²Rr², which comes from Pappus's centroid theorem: the volume of a solid of revolution equals the area of the rotated shape (here, πr², a circle) times the distance traveled by its centroid (2πR, the circumference of the central circle).

If r > R, the torus becomes 'self-intersecting' geometrically, meaning the tube would overlap itself at the center rather than forming a clean donut hole. The volume formula still computes a number, but it no longer represents a physically sensible donut shape; a valid torus requires R > r.

Surface area is A = 4π²Rr, also derived from Pappus's theorem but using the circle's circumference (2πr) instead of its area. Comparing the two formulas, volume uses r² (an area-based term) while surface area uses r (a length-based term), consistent with volume and area always differing by one dimension.

Yes, this standard formula assumes the tube's cross-section is a perfect circle; a torus with an elliptical or irregular tube shape would need a different, more complex calculation. Most practical torus shapes, like O-rings or donuts, are close enough to this ideal for the formula to be accurate.