Free Torus Volume Calculator

Understanding the Torus and Its Volume

The torus volume calculator is a practical tool designed to quickly determine the volume of a torus—the familiar doughnut or ring shape seen in bicycle tires, wedding bands, and even particle accelerators. By entering the torus’s two fundamental radii, users can obtain the volume in seconds using the well‑known torus volume formula. This donut volume calculator also serves as a ring volume calculator, making it useful for engineering, design, and mathematical exploration.

What Is a Torus?

A torus is a three‑dimensional surface generated by revolving a circle around an axis that lies in the same plane. The resulting shape resembles a donut, a tire, or a hoop. It features two key radii:

• Minor radius ($r$): the radius of the revolving circle (the cross‑section thickness).
• Major radius ($R$): the distance from the center of the torus to the center of the cross‑section.

Depending on the relationship between these radii, a torus can take three forms:

• Ring torus ($R > r$) – the most common, smooth donut shape.
• Horn torus ($R = r$) – a degenerate case where the inner radius shrinks to zero.
• Spindle torus ($R < r$) – a self‑intersecting shape; not covered by this calculator.

Often, the torus is described by its inner radius $a$ and outer radius $b$. These are derived from the two principal radii:

$$a = R - r, \qquad b = R + r.$$

Volume Formulas for a Torus

The volume of a torus can be expressed in two equivalent forms:

1. Using the minor and major radii:

$$V = 2\pi^{2} r^{2} R.$$

2. Using the inner and outer radii:

$$V = 0.25\pi^{2} (b - a)^{2} (b + a).$$

The second formula is what the torus volume calculator employs. The relationship between the two sets of radii is:

$$r = \frac{b - a}{2}, \qquad R = \frac{a + b}{2}.$$

Important notes:

• The calculator is intended for ring‑type and horn‑type tori only. Spindle tori ($R < r$) are not supported.
• For a horn torus, $R = r$ and consequently the inner radius $a = 0$. The formulas remain valid in this scenario.
• The result is given in cubic units consistent with the input length units (e.g., mm³, cm³, m³).

How to Use the Torus Volume Calculator

Using the donut volume calculator is straightforward and requires only two inputs:

1. Inner radius ($a$) – the distance from the center of the torus hole to the inner surface.
2. Outer radius ($b$) – the distance from the center to the outermost edge.

Once both values are entered, the calculator automatically applies the torus volume formula and displays the result. No manual unit conversion is needed within the tool.

Example Calculation

Let’s work through a typical example. Suppose a torus has a cross‑section radius $r = 40\ \text{mm}$ and a major radius $R = 100\ \text{mm}$.

First, convert to inner and outer radii:

$$a = R - r = 60\ \text{mm},\qquad b = R + r = 140\ \text{mm}.$$

Now apply the formula:

$$V = 0.25\pi^{2} (b - a)^{2} (b + a) = 0.25\pi^{2} (140 - 60)^{2} (140 + 60) \approx 3\,158\,273\ \text{mm}^{3}.$$

The result is about 3.16 million cubic millimeters, which can be converted to other units using a volume converter. This demonstrates how the torus capacity calculator quickly produces a concrete number, whether you’re designing a ring, sizing a donut‑shaped tank, or checking material for a toroidal component.

If you also need the surface area of the same torus—or its surface‑area‑to‑volume ratio—dedicated calculators are available for those tasks.