Understand the ring torus

A ring torus is formed by rotating a circle around a coplanar axis that does not meet it. The major radius R measures from the axis to the tube centre. The tube radius r measures the circular cross-section, with R > r.

Properties that make the calculations work

Its outer radius is R+r and its hole radius is R−r. Its volume is the tube’s circular area multiplied by the distance travelled by its centre. The analogous product of circumferences gives surface area.

Where the model is useful

Use major and tube radii, outside and hole radii, or inverse area and volume tools. These calculators cover an ideal ring torus only; horn and self-intersecting spindle tori use different geometric interpretations.

Measure the right thing

Major radius is not outer radius. All tools reject R ≤ r so that the result remains a ring torus.

Sketch the figure and mark every known measurement before choosing a calculator. A length measured along an edge is not automatically a perpendicular height. If a formula needs a radius, check that the value has not been measured across a full diameter. Write the unit beside each known quantity before entering it.

Scale and units

Multiplying every length by k multiplies any area by k² and any volume by k³. This is a useful check: doubling all dimensions cannot merely double an area. Inputs expressed in centimetres produce cm² for area and cm³ for volume; the unit selector relabels rather than converts entered numbers.

Continue learning

Wolfram MathWorld: Ring torus provides reference reading. Our summaries and examples are written for this portal and are not a specialist-reviewed textbook. Try a worked example or suggest a correction.