Torus Volume Calculator
Calculate the volume of a torus (donut shape) using its major and minor radii instantly.
Enter the major radius (R) and minor radius (r) of the torus, then click Calculate to get the volume.
Torus Volume Calculator
Calculate the volume of a torus (donut shape) using its major and minor radii instantly.
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 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 handles all positive values of R and r. When r equals R the torus is a horn torus (the inner hole closes to a point), and the formula V = 2π²Rr² still applies correctly. When r exceeds R the shape becomes a spindle torus whose surfaces self-intersect; the mathematical volume is still V = 2π²Rr² but physical interpretation requires care. All results are dimensionless in unit terms: enter measurements in meters to get cubic meters, in centimeters to get 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
- Identify the major radius R — the distance from the center of the torus to the center of the tube.
- Identify the minor radius r — the radius of the circular cross-section of the tube.
- Enter both values in the corresponding input fields using consistent units.
- Click Calculate Volume. The result is shown immediately in cubic units matching your input.
- 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?
Mathematically the formula V = 2π²Rr² remains valid, but the resulting shape is a spindle torus whose inner surfaces overlap and self-intersect. For engineering purposes (O-rings, tubes, rings) you almost always need r < R to represent a physically realizable ring shape.
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).