Look a little closer.

A circle swept around an outside axis. These connections help explain why the figure behaves as it does, beyond simply entering numbers into a formula.

01 / A useful relationship

The volume follows Pappus’s centroid theorem.

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.

02 / A different perspective

A ring torus has one central hole and no boundary edges.

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.

03 / A connection to explore

Keeping the tube radius fixed makes volume and surface area proportional to major radius.

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.

Try the idea with numbers

Open the major and tube radii calculator, note its outputs, then multiply each length input by two. Volume should increase by eight and surface area by four, when every relevant length is doubled. Angle inputs and dimensionless ratios remain unchanged. This scaling experiment checks the structure of the formula rather than the precision of any real measurement.

Further reading: Ring torus at Wolfram MathWorld.