How to Use the Torus Calculator

This calculator allows you to compute various measurements of a torus (or toroid) given known values.

To compute the surface area and volume of a torus in terms of it's two radii, click the 'Dimensions' button and enter the values for the major radius and the minor radius. The surface area and volume will appear in the both the torus preview and the math area, where the formulae used to make the calculations are shown. To save either the calculated surface area or volume, click the button that shows the resulting value and it will be copied to the clipboard, or click the 'Save' button to store the calculation in the calculator tape for later reference.

To calculate one of the torus's radii and volume in terms of the torus surface area, you can click the 'Area' button and supply the known values for the area. This will also display buttons with the missing torus radius and volume values that can be used to copy these calculations to the clipboard.

Finally, to calculate one of the torus's radii and area in terms of the torus's volume, click the 'Volume' button and enter the known volume.

What is a Torus?

A torus is a curved symetrical three dimensional shape that you might recognize as a donut or a bagel. You might also find torus shaped objects in the form of inner tubes or life bouys. The word torus comes from Latin where it is used generically to refer to a bump or protrudence.

You create a torus by sweeping a smaller circle perpendicularly around a path defined by a larger circle, and as such the torus is defined mathemetically by the radius of the larger circle (the major radius) and the smaller circle (the minor radius.) These two radii are shown in this torus calculator as the variables "R" and "r" respectively.

A torus has a number of interesting applications in electromagnetic phenomena, and you'll frequently encounter this shape in things like toroid magnets, chokes or other electronic components. Another application is in the structure of a prototype fusion reactor design called the tokamak, one of which is the ITER reactor being constructed in France. In this application, a torus shaped chamber is used to contain very high energy plasma in a torus shaped magnetic field to attempt to produce a sustained fusion reaction (and hopefully very inexpensive electric power.)

Mathematical Properties of Tori

A torus can be considered a cylinder bent into a closed circular shape, and the math associated with the torus share some of the elegance with cylinders and circles.

Like a cylinder, is a volume defined by a radius of a circle extended over a length. In the case of a cylinder, that length is the height. For a torus, that length is the outer circle's circumference. If you were to cut a torus and straighten it out into a cylinder, you probably have some instinct that the volume of the torus is going to be similar to the volume of a cylinder. However, what about the torus's inside edge? That portion of the torus's volume is going to be "compressed" relative to our straightened out cylinder shape. Similarly, the torus's outer edge is going to contain volume that is "stretched" compared to our straightened cylindrical shape. It turns out the amount of stretching and compressing balances out perfectly, so we can take the equations for circle circumference and for cylinder volume...

Equation for Area of a Circle

$$C_{\mathrm{circle}}=2\pi r$$

Equation for Volume of a Cylinder

$$V_{\mathrm{cylinder}}=\pi r^{2}h$$

...and substite the circumference for cylinder height to get the equation for the volume of a torus...

Equation for Volume of a Torus

$$V_{\mathrm{torus}}=\pi r^2(2\pi R)$$

$$V_{\mathrm{torus}}=2{\pi }^{2}R{r}^2$$

Unfortunately, the area of a torus doesn't have this same relationship. The equation of the area of a torus is still very elegant however...

Equation for Area of a Torus

$$V_{\mathrm{torus}}=4{\pi }^{2}Rr$$

Torus Calculator Definitions

Torus

A torus is a three dimensional surface described by a two radii to form a shape similar to a bagel or a donut. The major radius defines the larger diameter of the torus around right a perpendicular circle is swept to form a donut shaped geometry. The smaller radius, called the torus's minor radius, defines the size of the donut's tube. This torus calculator uses the variable R to denote the major radius and the variable r to renote the minor radius. The word torus can refer to the shape's surface, or the volume it encloses.

Tori

The plural of the word torus is tori.

Toroid

A synonym for a torus.

Horn Torus

When a torus has a major radius and a minor radius that equal, there is no central hole in the torus. This special case is called a horn torus. While this torus calculator finds the surface area and volume of ring tori, a the calculator's results for a horn torus are equivalent.

Major Radius

The major radius, identified in this torus calculator as the value R, defines the circlar path around which a circle defined by the minor radius is swept to define the torus volume. The major radius is shown in this torus calculator and the equations in this decription as the variable R.

Minor Radius

The minor radius, identified in this torus calculator as the value r, defines the circle swept around a the major radius to create the torus. The minor radius is shown in this torus calculator and the equations in this decription as the variable r.

Ring Torus

When you imagine a traditional torus shape similar to a donut, the shape has a hole through the middle, with that hole being defined by the difference between the major radius and the minor radius. This torus calculator finds the surface area and volume of a ring torus.

Spindle Torus

When a torus has a minor radius greater than the major radius, not only is there no central hole in the torus, but there is volume overlap as the circle defined by the minor radius is swept around the outer circle. A torus with these conditions is called a spindle torus. While this torus calculator does not calculator area of volume for a spingle torus and if you enter radii that would create a spindle torus it will ask you to adjust the values accordingly.

Toroidal

When something is shaped like a torus, it can be described as being toroidal.

Torus Calculator Updates

9 September 2018

Initial version of the torus calculator.

10 September 2018

Added calculator descriptions of toroid terms.