Calculate the volume, surface area, cross-section area, tube circumference, and outer/inner diameter of a torus from its major and minor radii. Includes presets for donuts, tires, and rings.
A torus is a doughnut-shaped surface of revolution generated by rotating a circle around an axis that lies in the same plane as the circle but does not intersect it. It is defined by two radii: the major radius R (center of the torus to the center of the tube) and the minor radius r (radius of the tube itself).
The volume of a torus is V = 2π²Rr², and the surface area is SA = 4π²Rr. These elegant formulas follow from Pappus' centroid theorem: the volume of a solid of revolution equals the cross-sectional area times the distance traveled by its centroid. Since the cross-section is a circle of area πr² and the centroid travels a distance of 2πR, V = πr² × 2πR = 2π²Rr².
Torus shapes appear everywhere in the real world: donuts, bagels, inner tubes, O-rings, tire tubes, and lifebuoys. In mathematics, the torus is a fundamental topological surface — it has genus 1 (one "hole") and can be constructed by identifying opposite edges of a rectangle.
This calculator takes the major radius R and minor radius r as inputs and computes volume, surface area, cross-sectional area of the tube, outer and inner diameters, tube circumference, and the volume-to-surface-area ratio. Multiple input options (radii, diameters, or outer/inner diameters) make it flexible. Presets for everyday torus-shaped objects and a scaling reference table round out the experience.
A torus is one of the few 3D shapes where the same object is often measured in several different ways. One problem might give major and minor radii, another might list diameters, and a product sheet may only provide outer and inner diameters. This calculator removes the conversion step and lets you work directly from the measurements you actually have.
That makes it useful for both teaching and practical estimation. You can compare volume against surface area, check whether a shape is a ring torus or a limiting horn torus, and see how strongly tube thickness controls the volume-to-surface-area ratio. It is a better fit than a bare formula when you need interpretation as well as arithmetic.
Volume: V = 2π²Rr² Surface Area: SA = 4π²Rr Cross-Section Area: A = πr² Tube Circumference: C_tube = 2πr Outer Diameter: D_outer = 2(R + r) Inner Diameter: D_inner = 2(R − r) Center Circumference: C_center = 2πR V/SA Ratio: r/2
Result: Volume ≈ 1,775.3, Surface Area ≈ 1,184.0
For R = 10, r = 3: Volume = 2π² × 10 × 9 ≈ 1,775.3. Surface Area = 4π² × 10 × 3 ≈ 1,184.0. Outer diameter = 2(10 + 3) = 26. Inner diameter = 2(10 − 3) = 14.
Torus problems are often expressed in whichever dimensions are easiest to measure. A machinist may know outer and inner diameters, a textbook may give R and r, and a catalog may specify diameters rather than radii. Converting all three views to the same internal geometry is the real first step, so a multi-mode calculator is useful because it prevents conversion mistakes before the volume formula is even applied.
The relationship between R and r determines the torus family. When R > r, the shape has a genuine hole and is called a ring torus. When R = r, the hole pinches to a point and you get a horn torus. When r > R, the surface self-intersects into a spindle torus. Seeing the classification next to the numbers is helpful because the same formula output can describe geometrically different shapes.
Volume grows with r², surface area grows with r, and the ratio V/SA = r/2 depends only on the tube radius. That means two tori with very different overall diameters can still share the same efficiency if their tube thickness matches. This is a useful perspective when comparing seals, tires, or decorative ring-like objects where wall thickness matters more than the overall ring diameter.
Volume = 2π²Rr², where R is the major radius (center to tube center) and r is the minor radius (tube radius). Use this as a practical reminder before finalizing the result.
The major radius R is the distance from the center of the torus to the center of the tube. The minor radius r is the radius of the tube itself.
When r = R, the inner hole shrinks to a point, creating a "horn torus". When r > R, the torus self-intersects (spindle torus).
Outer diameter = 2(R + r), inner diameter = 2(R − r). The inner diameter is positive only when R > r.
A torus is a surface of revolution of a circle. Pappus' theorem states that the volume equals the cross-sectional area (πr²) times the path length of its centroid (2πR).
Donuts, bagels, O-rings, inner tubes, lifebuoys, tires, tokamak fusion reactors, and particle accelerator rings are all torus-shaped. Keep this note short and outcome-focused for reuse.