Torus Volume Calculator
Calculate the volume of a torus (donut shape) using its major and minor radii instantly.
Calculate volume and surface area from two radii, or solve the minor radius from a major radius and volume.
Torus Volume Calculator
Calculate the volume of a torus (donut shape) using its major and minor radii instantly.
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About the Torus Volume Calculator
A torus is a surface of revolution generated by revolving a circle in three-dimensional space around an axis coplanar with the circle. When the axis does not intersect the circle itself, the result is a ring torus — the familiar donut shape seen in O-rings, tire inner tubes, decorative rings, and many engineering components. The volume enclosed by this surface has a formula that is clean and compact thanks to Pappus's centroid theorem.
The volume of a ring torus is V = 2π²Rr², where R is the major radius (the distance from the center of the torus to the center of the circular tube) and r is the minor radius (the radius of the tube's circular cross-section). This formula can be understood intuitively: the cross-sectional area of the tube is πr², and the tube travels a total path length of 2πR around the central axis, so by Pappus's theorem the volume is simply 2πR × πr² = 2π²Rr².
The volume formula differs from the surface area formula (SA = 4π²Rr) in that it contains r² rather than r, meaning volume grows proportionally with the square of the tube radius. Doubling the tube radius while holding R constant quadruples the volume but only doubles the surface area. This distinction matters significantly in engineering: if you double the wall thickness of a toroidal pipe, you need four times as much material by volume but only twice the outer coating area.
Practical applications of torus volume calculations span many fields. Mechanical engineers calculate the volume of O-rings and gaskets to determine their compressed size and sealing pressure. Chemical engineers compute the volume of toroidal reaction vessels and mixing chambers. Industrial designers use it when estimating the mass of ring-shaped castings or moldings from material density. Architects and structural engineers apply it to toroidal structural elements to compute material quantities and weights. Food scientists even use it to estimate the volume of annular baked goods.
The calculator classifies every positive pair of R and r. When R is greater than r, the shape is a ring torus and the standard formula applies. When r equals R, the torus is a horn torus and its inner hole closes to a point. When r exceeds R, the shape is a self-intersecting spindle torus. Pappus's ring-torus derivation does not apply in that case, so the calculator identifies the shape but deliberately suppresses the standard volume result. Enter measurements in meters to get square meters and cubic meters, in centimeters to get square centimeters and cubic centimeters, and so on.
Torus volume examples
Four worked examples applying the torus volume formula to real-world objects.
| Object | Volume | Details |
|---|---|---|
| Standard torus: R = 10, r = 3 | ≈ 5,583.1 cubic units | V = 2π² × 10 × 9 = 180π² ≈ 5,583.1. A medium torus with a relatively wide tube; typical of a ring-shaped structural element. |
| O-ring (thick): R = 5, r = 2 | ≈ 394.8 cubic units | V = 2π² × 5 × 4 = 40π² ≈ 394.8. A thick ring or O-ring where the tube radius is close to the major radius. |
| Large thin tube: R = 20, r = 2 | ≈ 1,579.1 cubic units | V = 2π² × 20 × 4 = 160π² ≈ 1,579.1. A large-diameter circular tube such as a bicycle or vehicle inner tube. |
| Decorative ring: R = 4, r = 1.5 | ≈ 177.7 cubic units | V = 2π² × 4 × 2.25 = 18π² ≈ 177.7. A small ring proportional to a decorative jewelry piece or a miniature donut. |
How to use the Torus Volume Calculator
- Choose whether to calculate from R and r or solve r from R and volume.
- Enter the requested positive values using consistent units.
- Click Calculate Volume to see the torus type and calculated measurements.
- For a spindle torus, use the classification warning instead of the standard ring-torus volume.
- Click Reset to clear the fields and start a new calculation.
Torus Volume Calculator FAQ
What is the formula for the volume of a torus?
The volume formula is V = 2π²Rr², where R is the major radius (center of torus to center of tube) and r is the minor radius (tube radius). This formula comes directly from Pappus's centroid theorem: the volume of a solid of revolution equals the cross-sectional area times the path length of the centroid, giving V = (πr²)(2πR) = 2π²Rr².
What is the difference between major radius R and minor radius r?
The major radius R measures how wide the overall torus ring is — it is the distance from the central axis of the torus to the midpoint of the tube. The minor radius r measures how thick the tube is — it is the radius of the circular cross-section. A donut with a large hole has a large R, while a plump donut with a small hole has r approaching R.
How does torus volume differ from torus surface area?
Volume (V = 2π²Rr²) measures the interior three-dimensional space in cubic units, used for capacity, mass, or material volume calculations. Surface area (SA = 4π²Rr) measures the outer skin in square units, used for coating, painting, or sealing area. Volume grows with r² while surface area grows linearly with r, so they scale differently when the tube thickness changes.
What units does the calculator output?
The output units are the cube of whatever unit you enter. Enter R and r in centimeters and the result is in cubic centimeters (cm³). Enter in meters and get cubic meters (m³). Enter in inches and get cubic inches (in³). No unit conversion is applied internally.
Can r be larger than R?
Yes, but the result is a spindle torus whose inner surfaces overlap and self-intersect. The standard V = 2π²Rr² formula comes from the ring-torus case and does not give the ordinary enclosed volume of a spindle torus, so this calculator suppresses that result. For engineering uses such as O-rings, tubes, and rings, R should be greater than r.
How do I find the volume of a hollow torus (torus-shaped tube with wall thickness)?
Calculate the volume of the outer torus using the outer minor radius (r_outer) and then subtract the volume of the inner torus using the inner minor radius (r_inner). Both calculations use the same major radius R. The wall volume = 2π²R(r_outer² − r_inner²).