Torus Calculator
Find a torus (ring) volume and surface area.
Result
How it is calculated
A torus is described by two radii: R, the distance from the central axis to the middle of the tube, and r, the radius of the tube itself. The tool computes V = 2π² · R · r² and A = 4π² · R · r. Both radii must be greater than zero and expressed in the same unit. The volume is in cubic units of that length and the surface area in square units; the tool does not add the unit for you. Results are rounded to two decimals. The tool does not check that R is larger than r. For a normal ring torus R should exceed r, otherwise the tube would overlap itself and the physical meaning of the numbers is lost.
Worked example
Use R = 5 and r = 2. Volume: 2π² × 5 × 2² = 2π² × 20 ≈ 394.78. Surface area: 4π² × 5 × 2 = 4π² × 10 ≈ 394.78. The two values match here only because 2 × R × r² and 4 × R × r are equal when r = 2; for other tube radii they differ and carry different units (cubic against square).
Common mistakes
- Entering diameters instead of radii, which makes the volume 8 times and the area 4 times too large.
- Using the outer radius of the ring as R; R is measured to the centre of the tube, not to its outer edge.
- Mixing units, such as R in centimetres and r in millimetres.
- Entering a tube radius larger than R, which describes a self-intersecting shape rather than a real ring.
- Reading the surface area number as cubic units or the volume as square units.
About this calculator
This tool calculates the volume and surface area of a torus — the donut-shaped surface formed by rotating a circle around an axis that doesn't intersect it — from two measurements: the center radius (the distance from the torus's central axis to the middle of the tube) and the tube radius (the radius of the circular cross-section itself). These two numbers alone fully determine the shape, since a torus is just a circle swept in a circular path.
It is used by students and engineers working through solid geometry problems that go beyond the basic sphere, cylinder and cone cases, and by people in mechanical design and manufacturing contexts — calculating the volume of a rubber O-ring or gasket, the material needed for a ring-shaped component, or the capacity of a torus-shaped tank — where the donut geometry shows up in the physical part being designed or specified.
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