Volume of a Torus
Calculate volume V = 2π²Rr² and surface area of a torus (donut shape).
Used in engineering tolerances, 3D modeling, and manufacturing ring-shaped components.
The Formula
V = 2π²Rr²
A torus is a donut-shaped solid formed by rotating a circle around an axis. Its volume depends on both the large radius (center to the middle of the tube) and the small radius (tube thickness).
Variables
| Symbol | Meaning |
|---|---|
| V | Volume of the torus |
| R | Major radius — distance from the center of the torus to the center of the tube |
| r | Minor radius — radius of the tube itself |
| π | Pi (approximately 3.14159) |
Example 1
A donut has R = 8 cm and r = 3 cm
V = 2π² × 8 × 3²
V = 2 × 9.8696 × 8 × 9
V ≈ 1,421.2 cm³
Example 2
A rubber O-ring has R = 25 mm and r = 2 mm
V = 2π² × 25 × 2²
V = 2 × 9.8696 × 25 × 4
V ≈ 1,974 mm³ ≈ 1.97 cm³
When to Use It
Use the torus volume formula when:
- Calculating the volume of O-rings, gaskets, and seals
- Designing donut-shaped structures or containers
- 3D modeling and printing ring-shaped objects
- Estimating material for toroidal shapes in engineering
Key Notes
- The formula requires R > r — when R = r the hole disappears (horn torus) and when R < r the shape self-intersects (spindle torus); both are degenerate cases where the standard formula is not accurate
- The volume can be understood via Pappus's centroid theorem: V = (2πR) × (πr²) — the circumference of the center circle × the area of the circular cross-section
- Surface area of a torus = 4π²Rr — notice volume grows as r² while surface area grows as r, so large-r tori pack more volume relative to surface area
- O-ring seals are compressed so their cross-section becomes slightly elliptical — the resulting contact pressure depends on the volume change, making precise r measurement critical for sealing performance