Torus Calculator
Calculate the volume, surface area, and other properties of a torus (donut shape).
A torus is a donut shape. R = distance from center to tube center. r = tube radius. Require r < R.
What Is the Torus Calculator?
The Torus Calculator computes the volume and surface area of a torus (donut shape). Enter the major radius R (distance from the center of the torus to the center of the tube) and the minor radius r (radius of the tube itself) to get precise geometric measurements.
Formula
How to Use
Enter the major radius R and the minor radius r of the torus. Ensure R ≥ r to get a valid ring torus (if R = r it becomes a horn torus, and R < r gives a spindle torus). The calculator instantly outputs volume, surface area, and the inner and outer diameters.
Example Calculation
Donut with R=5 cm, r=2 cm: Volume = 2π²×5×4 = 40π² ≈ 394.8 cm³. Surface Area = 4π²×5×2 = 40π² ≈ 394.8 cm². Note: for a torus, V and SA happen to have the same formula but different units.
Understanding Torus
A torus is the surface of revolution generated by revolving a circle of radius r in three-dimensional space about an axis coplanar with the circle, at a distance R from its center. The result is a donut-shaped surface that is fundamental in differential geometry, topology, and physics.
The volume and surface area formulas are elegantly derived using Pappus's centroid theorem: the volume equals the area of the cross-sectional circle (πr²) times the distance traveled by its centroid (2πR), giving V = 2π²Rr². Similarly, the surface area equals the circumference of the circle (2πr) times the same centroid distance, giving SA = 4π²Rr.
Toroidal geometry is critically important in plasma physics: the tokamak reactor design uses a toroidal magnetic field to confine plasma for nuclear fusion. Tori also appear in antenna design (toroidal coils), mechanical sealing (O-rings), and as the fundamental domain in certain topological constructions. In mathematics, the torus is a genus-1 surface and serves as the canonical example of a non-simply-connected surface.
Frequently Asked Questions
What is the difference between R and r in a torus?
R (major radius) is the distance from the center of the torus to the center of the circular tube. r (minor radius) is the radius of the tube itself. Together they define the shape: R controls the overall size and r controls the tube thickness.
What is a ring torus vs horn torus vs spindle torus?
Ring torus: R > r (hole in the middle — standard donut shape). Horn torus: R = r (inner circle collapses to a point). Spindle torus: R < r (self-intersecting shape with no hole).
Where do tori appear in real life?
Tori appear as donut shapes, O-rings, inner tubes, life rings, some magnetic fusion reactor designs (tokamaks), atmospheric vortex rings, and in topology (the torus is a fundamental surface).
Is the surface area formula exact?
Yes. The formula SA = 4π²Rr is exact for a perfect mathematical torus (generated by revolving a circle about an axis). Real manufactured tori (O-rings, etc.) approximate this.
Is this calculator free?
Yes, completely free with no account required.
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