Torus Calculator
Volume and surface area of a torus (donut).
Volume
1,776.5288
donut / ring torus
Surface area
1,184.3525
Outer diameter
26
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How the Torus Calculator works
The Torus Calculator instantly computes the volume and surface area of any torus (donut-shaped solid) given two radii, making it ideal for students, engineers, designers, and 3D printing enthusiasts working with ring-shaped geometry.
A torus is generated by revolving a circle of radius r (the tube radius) around an external axis at a distance R (the major radius) from the center of that circle. The result is the familiar donut or ring shape seen everywhere from architecture to plumbing fittings to bagels. Because it is a surface of revolution, its volume and surface area can be derived elegantly using Pappus's centroid theorem, which states that the volume of a solid of revolution equals the area of the shape being revolved multiplied by the distance traveled by its centroid.
The calculator requires exactly two inputs: the major radius R, which is the distance from the center of the torus to the center of the tube, and the minor radius r, which is the radius of the circular tube itself. A critical constraint is that R must be greater than or equal to r for a standard ring torus. When R equals r, the inner hole collapses to a point, creating a special case called a horn torus. When R is less than r, the shape self-intersects and becomes a spindle torus, which this calculator does not handle. Keeping R larger than r ensures a clean, hollowed ring shape with a visible hole in the center.
Volume is computed as V = 2π²Rr², which grows with both radii but is especially sensitive to the tube radius r because it appears as a square. Surface area is computed as A = 4π²Rr. Notice that surface area scales linearly with both R and r, while volume scales quadratically with r — so doubling the tube thickness more than doubles the material volume, a fact that catches many designers off guard when estimating print material or construction costs. Both formulas use π² (approximately 9.8696), not just π, because two circular dimensions are involved.
A common mistake is confusing R and r with diameter instead of radius — always halve a measured diameter before entering it. Another frequent error is using the outer radius of the whole torus as R; the major radius R is specifically measured to the centerline of the tube, not to the outer edge. If you know the outer radius (R_out) and inner radius (R_in) of a donut, convert using R = (R_out + R_in) / 2 and r = (R_out − R_in) / 2. Consistent units are essential: mixing centimeters and millimeters in the same calculation will produce wildly incorrect results.
Formula
V = 2π²Rr² · SA = 4π²Rr
Pro tips
- If you only know the outer and inner diameters of a torus, calculate R = (D_outer + D_inner) / 4 and r = (D_outer − D_inner) / 4 before entering values — dividing by 4 converts diameters to radii in one step.
- When estimating material for 3D printing, remember that volume scales with r², so even a small increase in tube radius significantly raises filament consumption; use the calculator to compare scenarios before committing.
- For architectural or engineering applications, verify that R ≥ r before running the calculation; if your design has R < r, you have a spindle torus and standard torus formulas no longer apply.
- Keep all measurements in the same unit throughout (e.g., all in centimeters); the resulting volume will be in cubic centimeters and surface area in square centimeters, which you can then convert as needed.
- To find the volume of a hollow torus shell (like a rubber O-ring with wall thickness t), calculate the volume of the full torus using r, then subtract the volume of the inner torus using r − t as the tube radius.
Key terms
- Major Radius (R)
- — The distance from the center of the torus to the center of the circular tube, determining how large the overall ring is.
- Minor Radius (r)
- — The radius of the circular cross-section of the tube, controlling how thick or fat the donut is.
- Ring Torus
- — The standard torus shape where R > r, resulting in a donut with a clearly visible hole through its center.
- Horn Torus
- — The special-case torus where R equals r, causing the inner hole to collapse to a single point.
- Pappus's Centroid Theorem
- — A geometric principle stating that the volume of a solid of revolution equals the revolved area times the distance traveled by the centroid, used to derive torus formulas.
- Surface of Revolution
- — A 3D shape formed by rotating a 2D curve or shape around an axis, of which the torus is a classic example.
