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
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
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
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"Torus Calculator" at https://MiniWebtool.com// from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: 2026-04-02
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