A torus is the shape of a donut, a bagel, an inner tube or an O-ring: a circle swept around an axis that lies in its own plane but outside it. Two numbers define it — the major radius R, from the center of the hole to the center of the tube, and the minor radius r, the radius of the tube itself. This calculator computes the volume and surface area from those, or from the easier-to-measure inner and outer radii, and lists the overall diameter, hole size, thickness and capacity.
How to use the torus calculator
- Choose whether you know the major and minor radius or the inner and outer radius.
- Enter the two values and pick the unit.
- Read the volume on the tape, with the surface area, both radii, the hole and outer diameters, the thickness, the aspect ratio and the capacity in liters and US gallons. The diagram shows the torus from above and in cross-section.
Torus formulas
Converting between the two sets of measurements:
| From inner radius b and outer radius a | From R and r |
|---|---|
| R = (a + b)/2 | inner radius = R − r |
| r = (a − b)/2 | outer radius = R + r |
Both formulas come from Pappus’s centroid theorems. The tube’s cross-section is a circle of area πr2 and circumference 2πr; its center travels a path of length 2πR. Multiply to get the volume (πr2 × 2πR) and the surface area (2πr × 2πR). In other words, a torus has the same volume as a cylinder of radius r and length 2πR — as if you cut the ring and straightened it.
Worked example
A foam swim ring has an outer radius of 13 cm and a hole radius of 7 cm.
Radii: R = (13 + 7)/2 = 10 cm and r = (13 − 7)/2 = 3 cm.
Volume: V = 2π2 × 10 × 32 = 180π2 ≈ 1,776.5 cm³, about 1.78 liters.
Surface area: A = 4π2 × 10 × 3 = 120π2 ≈ 1,184.4 cm².
Size: overall diameter 26 cm, hole diameter 14 cm, thickness 6 cm.
Types of tori
| Shape | Condition | Hole |
|---|---|---|
| Ring torus | r < R | yes — the familiar donut |
| Horn torus | r = R | shrinks to a single point |
| Spindle torus | r > R | none; the surface intersects itself |
A large ratio R/r gives a thin ring like a bracelet or an O-ring; a ratio near 1 gives a fat bagel with a tiny hole.
Practical uses
- O-rings and gaskets. The volume of the rubber ring, and therefore its weight, follows from the cross-section diameter (2r) and the ring’s centerline diameter (2R).
- Inner tubes and swim rings. The air volume matters for buoyancy and inflation; the surface area tells you how much material the tube needs.
- Toroidal tanks and coils. Fuel tanks shaped like rings and toroidal transformer cores use the same formulas for capacity and winding length.
- Food. A bagel’s dough volume, or the glaze needed to cover a donut, are classic torus estimates.
Keep units consistent: with lengths in centimeters, volume comes out in cubic centimeters (1,000 cm³ = 1 liter) and area in square centimeters.
Related tools
For a straight tube with the same cross-section, use the cylinder calculator or, for a hollow pipe, the tube calculator. The flat ring seen from above is covered by the annulus calculator, round solids by the sphere calculator, and other shapes by the volume calculator.
Frequently asked questions
What is the volume of a torus?
V = 2π²Rr², where R is the major radius (center of the hole to the center of the tube) and r is the minor radius (the tube's radius). A torus with R = 10 cm and r = 3 cm holds 180π² ≈ 1,776.5 cm³.
What is the surface area of a torus?
A = 4π²Rr. For R = 10 and r = 3 that is 120π² ≈ 1,184.4 square units.
How do I get R and r from measurements?
Measure the outer diameter and the hole diameter. R is the average of the two radii and r is half their difference: R = (outer + inner)/2 and r = (outer − inner)/2 when both are radii.
Why are these formulas so simple?
They follow from Pappus's centroid theorems: a solid swept by rotating a shape around an axis has volume equal to the shape's area times the distance its centroid travels. The tube's circle, area πr², travels a circle of length 2πR.
What if r is larger than R?
The surface passes through itself, forming a spindle torus with no hole, and the simple formulas overcount the overlap. This calculator covers ring tori (r < R) and the horn torus (r = R).