Torus Calculator
Calculate the volume and surface area of a torus from its major and minor radii.
This tool is for informational and educational purposes only. It is not a substitute for professional financial, medical, legal, or engineering advice. See Terms of Service.
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Request a ToolHow to Use the Torus Calculator
Enter the major radius (distance from the center of the torus to the center of the tube) and the minor radius (radius of the tube itself).
- Enter the major radius (R). This is the distance from the center of the torus to the center of the circular tube.
- Enter the minor radius (r). This is the radius of the tube. It must be less than R.
- Read the results. The volume is the primary result, with the surface area shown below.
Volume = 2 * pi^2 * R * r^2. Surface Area = 4 * pi^2 * R * r.
About Toruses
A torus is a donut-shaped surface of revolution generated by rotating a circle around an axis in its plane. The major radius R is the distance from the axis to the center of the revolving circle, and the minor radius r is the radius of that circle. Toruses appear in mathematics (topology), physics (magnetic confinement in tokamaks), architecture, and everyday objects like donuts, tires, and O-rings. A torus is only valid when r is less than R (otherwise the shape self-intersects).
Frequently Asked Questions
What is the volume of a torus?
V = 2 * pi^2 * R * r^2, where R is the major radius and r is the minor radius. This can also be written as (pi * r^2) * (2 * pi * R), the cross-sectional area times the path length.
What is the surface area of a torus?
SA = 4 * pi^2 * R * r. This equals the circumference of the tube (2*pi*r) times the distance traveled by the tube center (2*pi*R).
What is the difference between major and minor radius?
The major radius (R) is the distance from the center of the torus to the center of the tube. The minor radius (r) is the radius of the tube itself. Think of R as how far out the donut extends, and r as how thick it is.