Torus Volume CalculatorFree online math calculator — instant results, no sign-up
Calculate the volume of a ring torus.
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About the Torus Volume Calculator
The Torus Volume Calculator finds the volume of a ring torus, a doughnut-shaped solid with a circular tube. It uses the major radius and tube radius to measure the space occupied by the solid, excluding the central hole. With the default major radius of 5 and tube radius of 2, the volume is 394.7841760436 cubic units.
- Major radius
- The major radius is the distance from the torus's central axis to the center of its circular tube, measured in the same length unit as the tube radius.
- Tube radius
- The tube radius is the radius of the tube's circular cross-section, measured in the same length unit as the major radius.
Formula and method
V = (πr²)(2πR) = 2π²Rr²; with R = 5 and r = 2, V = 2π² × 5 × 2² = 40π² ≈ 394.7841760436.
- VVolume of the solid torus, in cubic units.
- RMajor radius: the distance from the central axis to the center of the tube.
- rTube radius: the radius of the circular cross-section.
- πPi, the ratio of a circle's circumference to its diameter.
Square the tube radius and multiply by π to obtain the tube's cross-sectional area. Multiply that area by 2π times the major radius, the circumference traced by the center of the tube. For the default inputs, this gives (π × 2²) × (2π × 5) = 40π² ≈ 394.7841760436 cubic units.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Major radius5
- Tube radius2
Results
- Result394.7841760436
How to use the Torus Volume Calculator
- 1Choose a single length unit for both radii.
- 2Enter the Major radius, using 5 for the default worked example.
- 3Enter the Tube radius, using 2 for the default worked example.
- 4Check that both radii are positive and that the major radius is greater than the tube radius, as required for a ring torus.
- 5Calculate the volume and read the Result in cubic units; the default inputs give 394.7841760436.
Reading your result
The Result row reports the volume occupied by the solid torus, not the volume of its central hole or its surface area. With Major radius = 5 and Tube radius = 2, Result = 394.7841760436 cubic units. If both radii are in centimeters, the result is in cubic centimeters; if both are in meters, it is in cubic meters.
Limitations
The formula assumes an ideal solid ring torus with a constant circular cross-section and requires R > r > 0. Equal radii produce a horn torus rather than a ring torus, while R < r produces a self-intersecting spindle torus for which this formula should not be used as an ordinary solid-volume calculation. Both inputs must use the same length unit. The calculator does not account for hollow tube walls, noncircular cross-sections, or manufacturing irregularities, and the displayed decimal is an approximation.
Common questions
Is the major radius measured to the outer edge of the torus?+
No. It is measured from the central axis to the center of the tube. The outer radius is R + r, and the radius of the central hole is R − r.
Does the calculated volume include the central hole?+
No. The result measures only the solid doughnut-shaped region. The empty space through the center is excluded.
Why is the tube radius squared but the major radius is not?+
The tube radius determines the cross-sectional area πr², while the major radius determines the centerline circumference 2πR. Multiplying these produces V = 2π²Rr².
Why do the default radii give 394.7841760436 rather than an exact decimal?+
With R = 5 and r = 2, the exact volume is 40π² cubic units. Its decimal expansion does not terminate, so 394.7841760436 is the rounded displayed value.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.