Torus Volume Calculator

The torus volume calculator will determine the volume of a torus for a given pair of radii. Ever wondered what the volume is of your doughnut or ring? The bicycle you ride also run because of a couple of tori. Read on to understand what is a torus and how to calculate the volume of torus.

A torus is a 3D shape obtained by revolving a circle around an axis. Check a similar revolving idea, but with a rectangle, in our right cylinder calculator. This shape is commonly found in doughnuts, rings, tires, tubes, etc. If you take a ring and circularly trace it around with a pencil, you get a torus. In modern design software, it is fairly easy to draw them by using a revolve command with a circle as a cross-section. A torus has two radii – the first radius is the radius of the cross-section r and the second radius is the distance between the center of the torus to the center of the cross-section R.

Based on the combinations of the two radii, we can obtain multiple types of tori. Such that:

• Ring type (R > r)
• Horn type (R = r)
• Spindle type (R < r)

Unlike the smooth, simple surface of a sphere, the spindle type starts to self-intersect. In addition to these radii, the torus can also be expressed in the form of two radii such as inner (a) and outer radii (b) of the torus. Mathematically, that is:

a = R - r
b = R + r

The volume V of the said torus is:
V = 2 * π² * r² * R
The volume can also be written in terms of the inner and outer radii:
V = 0.25 * π² * (b - a)² * (b + a)
The above equation is used in the calculator as the volume of torus formula where:

• r = (b - a) / 2
• R = (a + b) / 2

What about its surface area? Check the torus surface area calculator to find out! You can then calculate the surface area to volume ratio.

Note: This calculator is only applicable to ring-type or horn-type tori. Also, in the case of a horn-type torus, i.e., R = r, minor radius, a becomes zero.

Follow three simple steps to find the volume of a torus.

• Step 1: Enter the inner radius of torus, a.
• Step 2: Enter the outer radius of torus, b.
• The calculator will now use the volume of torus formula to return the result.

Find the volume of torus having the radius of cross-section as 40 mm, and the distance from the center of torus to the center of cross-section as 100 mm.

Given: radius of cross-section, r = 40 mm; radius of torus, R = 100 mm;

Let us first convert the radii into the inner and outer radius, a and b.

a = R - r = 60 mm
b = R + r = 140 mm

• Step 1: Enter the inner radius of torus, a = 60 mm.
• Step 2: Enter the outer radius of torus, b = 140 mm.
• Step 3: Using volume of torus formula,

V = 0.25 * π²> * (b - a)² * (b + a)
V = 0.25 * π² * (140 - 60)² * (140 + 60) = 3158273 mm³
You can convert this large number to a different unit using our volume converter.
If you are done with the volume of a torus, hop over to the volume calculator to find this quantity for other various shapes, such as a capsule or a conical frustum.