Torus Volume Calculator
Enter the major radius (center of tube to center of torus) and the minor radius (tube radius) to instantly compute the volume and surface area of your torus. Switch between metric and imperial units, pick any input mode (R and r, or inner and outer radii), and see the full step-by-step working below the result.
Formula
Worked example
A torus with major radius R = 10 cm and tube radius r = 3 cm: V = 2 * pi^2 * 10 * 9 = 1776.53 cm^3. Surface area SA = 4 * pi^2 * 10 * 3 = 1184.35 cm^2. Inner radius a = 10 - 3 = 7 cm, outer radius b = 10 + 3 = 13 cm.
What is a torus?
A torus is the three-dimensional surface formed by rotating a circle around an axis that lies in the same plane but does not intersect the circle. The result looks like a doughnut, an O-ring, a tire inner tube, or a life preserver. It is described by two measurements: the major radius R (from the center of the torus to the center of the circular tube) and the minor radius r (the radius of the tube itself). Equivalently you can describe it by the inner radius a (the nearest edge from the axis) and the outer radius b (the farthest edge), where R = (a + b) / 2 and r = (b - a) / 2.
Torus volume and surface area formulas
Both formulas come from Pappus's centroid theorem, which states that the volume of a solid of revolution equals the area of the cross-section multiplied by the distance traveled by its centroid. For a torus, the cross-section is a circle of area pi * r^2 and the centroid travels a circular path of circumference 2 * pi * R, giving V = 2 * pi^2 * R * r^2. By an analogous argument for the surface, SA = 4 * pi^2 * R * r. In terms of inner and outer radii: V = 0.25 * pi^2 * (b - a)^2 * (b + a) and SA = pi^2 * (b^2 - a^2).
The three types of torus
Mathematicians classify a torus by the relationship between R and r. When R is greater than r the torus is a ring torus: it has a visible hole in the center, and this is the familiar doughnut shape. When R equals r the tube just touches the central axis at a single point, creating a horn torus with no central hole but also no self-intersection. When R is less than r the tube overlaps itself through the central axis, producing a spindle torus whose outer surface is self-intersecting. Only the ring and horn cases are physically realizable as solid objects. At the extreme, if R equals zero the torus degenerates into a sphere of radius r.
Real-world applications
Torus geometry appears throughout engineering and design. O-rings and shaft seals are toroidal, and their volume determines how much material is needed. Tire inner tubes, inflatable life rings, and rubber gaskets are all ring tori. In architecture, domed roofs and arches often use toroidal sections. Particle accelerators like cyclotrons and tokamak fusion reactors use a toroidal magnetic field geometry. In mathematics, the torus is a fundamental surface in topology: it has genus one (one hole), and its surface can be described by two angle coordinates, making it essential in complex analysis and the study of differential equations.
Torus types by R and r relationship
| Condition | Torus type | Description | Example |
|---|---|---|---|
| R > r | Ring torus | Hole visible in the center; most common shape | Doughnut, O-ring, life ring |
| R = r | Horn torus | Tube meets the axis at exactly one point; no central hole | Pinched inner ring |
| R < r | Spindle torus | Tube passes through the axis; self-intersecting surface | Theoretical / degenerate |
| R = 0 | Sphere | Degenerate case: torus collapses to a sphere of radius r | Ball, sphere |
A torus is classified by comparing the major radius R (center-to-tube-center) with the minor radius r (tube radius).
Frequently asked questions
What is the formula for the volume of a torus?
The volume is V = 2 * pi^2 * R * r^2, where R is the major radius (center of torus to center of tube) and r is the minor radius (tube radius). Equivalently, if you know the inner radius a and outer radius b, you can write V = (pi^2 / 4) * (b - a)^2 * (b + a). Both formulas derive from Pappus's centroid theorem.
What is the surface area of a torus?
The surface area is SA = 4 * pi^2 * R * r. In terms of inner and outer radii, SA = pi^2 * (b^2 - a^2). This formula counts only the outer (or inner) surface of the torus, not any cut cross-sections.
What is the difference between the major radius and the minor radius?
The major radius R is the distance from the center of the hole to the center of the tube - in a doughnut this is roughly the radius of the doughnut itself. The minor radius r is the radius of the circular tube cross-section - in a doughnut this is roughly half the thickness of the dough. Together they fully describe the shape.
Can I use inner and outer radii instead of R and r?
Yes. Select the "Inner radius a + outer radius b" input mode. The calculator derives R = (a + b) / 2 and r = (b - a) / 2 before computing. This mode is useful when you can measure the torus with a ruler or calipers from the outside.
What units does this calculator use?
Choose metric (centimeters) or imperial (inches) from the units selector. Volume is then reported in cm^3 or in^3 respectively, and surface area in cm^2 or in^2. For any other unit (mm, m, ft), enter values in that unit and interpret the result in the corresponding cubic or square unit.
What is a ring torus, horn torus, and spindle torus?
A ring torus (R greater than r) has a visible central hole - this is the everyday doughnut shape. A horn torus (R equals r) has no central hole because the inner edge of the tube just touches the axis. A spindle torus (R less than r) has the tube looping through itself; it is self-intersecting and not physically realizable as a solid shell.
How do I find the volume of a hollow torus (e.g. a rubber O-ring with wall thickness t)?
Model the outer torus with minor radius r_out and the inner cavity as a torus with the same R but minor radius r_in = r_out - t. The material volume is V_out minus V_in = 2 * pi^2 * R * (r_out^2 - r_in^2).