Torus Volume and Surface Area Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The volume of a torus is calculated with V = 2π² × R × r², where R is the major radius (from the center of the torus to the center of the tube) and r the tube radius. For R = 10 cm and r = 3 cm, the volume is about 1776.53 cm³.
Explanation
A torus is the surface generated by rotating a circle (the "tube") around an axis outside that circle — the geometric shape of an inner tube, a donut, or a life ring. Its volume and surface formulas follow from Pappus's theorem: when a plane figure rotates completely around an external axis, the swept volume equals the area of that figure multiplied by the distance traveled by its centroid (2πR, the circumference of the circle described by the center of the tube). Applying this principle to a disk of radius r (area πr², as in the circle area and perimeter calculator) rotating at distance R from its center gives V = πr² × 2πR = 2π²Rr² directly; the same reasoning applied to the circle's perimeter (2πr) rather than its area gives the surface formula, S = 2πr × 2πR = 4π²Rr. Unlike a sphere, whose volume depends on a single radius, a torus has a shape defined by two independent radii: the ratio between R and r determines whether the torus looks like a thick life ring (R close to r) or a thin ring (R much larger than r).
Example: major radius 10 cm, tube radius 3 cm
Inputs
Major radius (R): 10 cm. Tube radius (r): 3 cm.
Calculation
V = 2π² × 10 × 3² = 2π² × 10 × 9 = 180π² ≈ 1776.53 cm³. S = 4π² × 10 × 3 = 120π² ≈ 1184.35 cm².
Result
This torus has a volume of about 1776.53 cm³ and a surface area of about 1184.35 cm².
Frequently asked questions
What happens if the major radius is smaller than the tube radius?
Geometrically, if R < r, the tube would pass through itself as it rotates, which no longer corresponds to a "classic" ring-shaped torus (this is called a self-intersecting or horn torus): the formulas in this calculator remain numerically valid, but the resulting figure no longer has the intuitive shape of a ring.
Why does π appear squared in these formulas?
Because the torus combines two distinct circular rotations: that of the small circle (the tube) on itself, and that of the whole tube around the main axis. Each of these two circularities contributes a factor of π to the calculation, hence the π² that does not appear in the formulas for a sphere or a cylinder, solids with a single rotation.
Does this formula apply to a torus with a non-circular cross-section?
No, it assumes a tube with a perfectly circular cross-section, the standard case of Pappus's theorem applied to a disk. A tube with an elliptical or any other cross-section would follow the same general principle (cross-section area × distance traveled by its centroid), but with a different area formula for the cross-section, and therefore a different numerical result.