The Donut-Shaped Torus

A torus resembles a donut or an inner tube. It is generated by revolving a circle of radius r about an axis that is coplanar with the circle but does not intersect it. The distance from the center of the circle to the axis of rotation is known as the major radius R. When R greatly exceeds r, the torus looks like a thin ring; when the two radii are comparable, it becomes plump like a bagel. Determining the volume and surface area of this shape is important in mechanical design, architecture, and computer graphics. For instance, O‑rings, inflatable tubes, and certain magnetic coils all take the form of a torus.

Mathematical Formulas

The volume and surface area of a torus can be derived using the Pappus centroid theorem or direct integration. In MathML, the formulas appear as:

and

These compact expressions reveal that both quantities scale linearly with the major radius and quadratically with the minor radius. The factor of \(2\pi\) accounts for the revolution of the generating circle, while the second \(\pi\) emerges from integrating the circumference of the circle itself. When either radius approaches zero, the torus degenerates into a circle or cylinder, and the formulas reduce accordingly.

Using the Calculator

Begin by measuring or specifying the major radius — the distance from the center of the tube to the center of the hole — and the minor radius, the radius of the tube. Enter both values in consistent units (meters, inches, or any other). After pressing Calculate, the script multiplies them according to the formulas above to produce the volume and surface area. The results appear instantly without any network connection. You can copy the output using the provided button.

Typical Dimensions

The table below lists sample torus sizes found in everyday objects:

Object	Major Radius (cm)	Minor Radius (cm)
Small O-Ring	1.5	0.2
Bagel	5	1.5
Pool Inner Tube	25	4

Worked Example

Suppose you design a rubber gasket with R = 3 cm and r = 0.5 cm. The volume becomes:

Evaluating yields roughly 14.8 cm³. The surface area is:

or about 59.2 cm². These calculations help you estimate the amount of rubber needed and the exposed surface for coatings or heat transfer.

Torus Geometry in Context

Beyond simple volume and area, the torus has fascinating mathematical properties. Its surface is a genus 1 manifold, meaning it contains a single "hole." In topology, tori serve as fundamental examples of surfaces with nontrivial loops. Many physical systems adopt toroidal shapes: magnetic confinement devices for fusion experiments, high-power toroidal transformers, and certain flotation devices. By understanding the basic geometry, engineers can tailor these devices for efficiency and performance.

Limitations of the Formula

The formulas assume a perfect torus with uniform radii and negligible material thickness. In practice, an O‑ring might have a slightly oval cross-section or varying diameter due to manufacturing tolerances. If precision is critical, 3D modeling software or experimental measurements may be required. Nevertheless, these equations provide an excellent approximation for design work and educational purposes.

Further Exploration

You can extend the concept of a torus to more exotic shapes. A horn torus forms when r equals R, resulting in the inner radius collapsing to zero. When r exceeds R, the surface self‑intersects, creating a spindle torus. Exploring these variations can lead to striking designs in art and architecture. The same formulas apply, though the physical interpretation changes as the hole shrinks or disappears.

Conclusion

This calculator offers a quick, offline method to evaluate a torus. Whether you are printing a 3D model, estimating fluid volume in a tube, or teaching geometry, simply input the major and minor radii to obtain accurate results. Torus shapes show up across science, engineering, and everyday items. Understanding their basic measurements unlocks a deeper appreciation for how geometry shapes the world around us.