Torus Volume Calculator
Calculate torus volume and surface area from major radius (R) and minor tube radius (r). Sized for O-rings, donuts, and toroidal tanks.
The area of the circular slice of the tube itself.
By Pappus's Centroid Theorem, the torus is generated by sweeping the circular cross section along this circular path.
The total internal volume of the ring torus or O-ring.
The volume of a ring torus equals 2 × π² × R × r², where R is the major radius (distance from center of hole to center of tube) and r is the minor radius (tube radius). For R = 6 in and r = 2 in, volume is 473.74 cubic inches (2.05 US gallons).
Torus Volume Calculator Formula
| Variable | Definition | Standard Units |
|---|---|---|
| V | Internal volume of the torus ring | in³, ft³, cm³, or m³ |
| R | Major radius: distance from center of hole to centerline of tube | in, ft, cm, or m |
| r | Minor radius: cross-sectional radius of the tube itself | in, ft, cm, or m |
Pappus’s Centroid Theorem and Torus Geometry
A ring torus is formed by revolving a circular disk of radius r along a circular path of radius R. By Pappus’s Centroid Theorem, the volume generated equals the cross-sectional area of the circle (A = πr²) multiplied by the distance traveled by its center of mass (C = 2πR). Multiplying (πr²) × (2πR) gives V = 2π²Rr².
Common Calculation Mistakes to Avoid
- Confusing major radius R with the outer overall diameter
- Setting minor radius r greater than major radius R for a ring torus
Frequently Asked Questions
What is the formula for the volume of a torus?▼
What is the surface area of a torus?▼
How do you calculate the volume of an O-ring rubber seal?▼
What is the difference between a ring torus, horn torus, and spindle torus?▼
Authoritative Sources & Reference Standards
- Guide for the Use of the International System of Units (SI), NIST Special Publication 811 — National Institute of Standards and Technology (NIST). Verified 2026-10-01.
Official SI unit definitions, conversion factors (Appendix B), and precision standards for cubic meters, liters, and US customary units.
- Common Core State Standards for Mathematics: High School Geometry (Geometric Measurement and Dimension HSG.GMD.A.1–3) — National Governors Association Center for Best Practices & Council of Chief State School Officers. Verified 2026-10-01.
Curricular standards for volume formulas of cylinders, pyramids, cones, and spheres, including Cavalieri’s principle.