Torus Calculator

Calculate the volume, surface area, and geometric properties of a torus (donut shape). Enter the major radius (R) and minor radius (r) to get instant results with step-by-step formulas and an interactive 3D cross-section diagram.

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About Torus Calculator

The Torus Calculator computes the volume, surface area, and geometric properties of a torus — a 3D donut-shaped surface of revolution. A torus is generated by revolving a circle of radius r (the minor radius, or tube radius) around an axis at distance R (the major radius) from the circle's center. Enter the major and minor radii to get instant results with step-by-step formulas and an interactive cross-section diagram.

Three Types of Torus

🍩

Ring Torus

R > r

Classic donut shape with a visible hole in the center. The most common type in everyday life.

📯

Horn Torus

R = r

The tube just touches itself at the center — there is no hole, but no self-intersection either.

🌀

Spindle Torus

R < r

The tube overlaps itself, creating a self-intersecting surface with no hole.

Key Formulas for a Torus

For a torus with major radius R (center of torus to center of tube) and minor radius r (radius of the tube):

Property	Formula	Description
Volume	\(V = 2\pi^2 R r^2\)	Enclosed 3D space
Surface Area	\(A = 4\pi^2 R r\)	Total outer surface
Outer Radius	\(R_{\text{outer}} = R + r\)	Center of torus to outermost point
Inner Radius	\(R_{\text{inner}} = R - r\)	Center of torus to hole edge
V/A Ratio	\(\frac{V}{A} = \frac{r}{2}\)	Depends only on tube radius

Real-World Applications

🍩

Donuts & Bagels

Classic torus-shaped food items

🛞

Tires & Inner Tubes

Toroidal shape for pneumatic tires

💍

Rings & Jewelry

Band rings follow torus geometry

🔬

Tokamak Reactors

Toroidal chamber for plasma fusion

🧲

Toroidal Coils

Inductors and transformers

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Pool Floats

Inflatable swim rings

Understanding Torus Geometry

A torus is mathematically defined as a surface of revolution: take a circle of radius r and revolve it around an axis that lies in the same plane as the circle but does not intersect it (for a ring torus). The distance from the axis to the center of the revolving circle is the major radius R. The parametric equations of a torus centered at the origin with the z-axis as its axis of symmetry are:

\(x = (R + r\cos\theta)\cos\phi\), \(y = (R + r\cos\theta)\sin\phi\), \(z = r\sin\theta\)

where \(\theta\) and \(\phi\) range from 0 to \(2\pi\). The volume formula \(V = 2\pi^2 R r^2\) can be derived using Pappus' theorem: the volume of a solid of revolution equals the area of the cross-section (\(\pi r^2\)) multiplied by the distance traveled by the centroid (\(2\pi R\)).

How to Use the Torus Calculator

Enter the major radius (R): Type the distance from the center of the torus to the center of the tube, or click a quick example like Donut, Tire, or Ring.
Enter the minor radius (r): Type the radius of the tube cross-section.
Click Calculate Torus: Press the button to compute all properties instantly.
Review the results: See volume, surface area, inner/outer radii, and other properties in the results cards. Use the diagram toggle buttons to show or hide dimensions, radii labels, and the axis of revolution.

Torus vs. Sphere vs. Cylinder

A sphere is a surface where every point is equidistant from the center — it has no hole. A cylinder has two flat circular ends connected by a straight surface. A torus has no flat faces and features a hole through the center (for ring tori). Topologically, a torus has genus 1 (one hole), while a sphere has genus 0. This fundamental difference means the Euler characteristic of a torus is 0 (versus 2 for a sphere), and its total Gaussian curvature integrates to 0 by the Gauss-Bonnet theorem.

FAQ

What is a torus?

A torus is a 3D surface of revolution shaped like a donut. It is generated by revolving a circle (with minor radius r) around an axis that is coplanar with the circle, at a distance R (major radius) from the center of the circle. The result is a ring-shaped surface with a hole in the center (when R > r).

How do you calculate the volume of a torus?

The volume of a torus is V = 2π²Rr², where R is the major radius (center of torus to center of tube) and r is the minor radius (radius of the tube). This formula comes from Pappus' centroid theorem: multiply the cross-sectional area (πr²) by the distance its centroid travels (2πR).

What is the surface area of a torus?

The surface area of a torus is A = 4π²Rr, where R is the major radius and r is the minor radius. This can also be derived from Pappus' theorem: multiply the circumference of the cross-section (2πr) by the distance its centroid travels (2πR).

What are the different types of torus?

There are three types: a ring torus (R > r) has a visible hole in the center like a donut, a horn torus (R = r) has the inner surface just touching itself at the center with no hole and no self-intersection, and a spindle torus (R < r) self-intersects because the tube is wider than the distance to the axis.

What is the difference between major radius and minor radius?

The major radius R is the distance from the center of the torus to the center of the tube cross-section. The minor radius r is the radius of the tube itself. Together they define the overall size and proportions of the torus. The ratio R/r determines the torus type and its visual appearance.

Reference this content, page, or tool as:

"Torus Calculator" at https://MiniWebtool.com/torus-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-02

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

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