Torus Volume Calculator
Calculate the volume of a torus (donut shape) from major and minor radii. Uses V = 2 x pi^2 x R x r^2. Perfect for engineering seals, rings, and toroids.
How to use this tool
- Enter major radius (r) and minor radius (r) in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your volume and the full breakdown beneath it.
Formula
V = 2 x pi^2 x R x r^2
How it works
Use V = 2π²Rr². R is the distance from the torus center to the tube center; r is the tube radius.
Worked example
Torus R=5, r=2
- R = 5, r = 2
- r^2 = 4
- V = 2 x pi^2 x 5 x 4 = 40 x pi^2
- V = 394.7841...
Volume = 394.7842 cubic units
Common mistakes to avoid
- Confusing the major radius R (distance from the torus centre to the centre of the tube) with the minor radius r (radius of the tube itself) — swapping them gives a completely different, incorrect volume.
- Forgetting that R must be greater than r; if r >= R the shape self-intersects and is no longer a standard torus.
- Using the formula for a hollow ring or pipe (a cylinder bent into a circle) which gives a different result from the true torus volume formula.
Key terms
- What is the major radius?
- The distance from the center of the torus to the center of the tube that wraps around it.
- What is the minor radius?
- The radius of the circular tube itself.
Frequently asked questions
- What is the difference between the major radius R and minor radius r?
- R is the distance from the centre of the torus (the hole) to the centre of the circular tube that forms the ring. r is the radius of that tube. For a standard doughnut, R is roughly the ring's overall radius and r is the thickness of the doughnut.
- What happens if r equals R?
- When r = R the inner hole disappears and the surface passes through the axis of symmetry, creating a horn torus. The volume formula V = 2*pi^2*R*r^2 still gives a mathematically valid result at this boundary.
- What are typical engineering applications of the torus volume calculation?
- Torus volumes are used when sizing O-ring seals (estimating elastomer volume), designing toroidal fuel tanks, and modelling circular magnetic cores (toroids) for inductors and transformers.