Torus Volume: Calculating the Space Inside a Donut Shape
Calculate torus volume using V = 2π²Rr². Learn Pappus's theorem, worked examples, and applications in engineering and physics.
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The volume of a torus — the space enclosed by the donut-shaped surface — is a fundamental calculation in geometry and engineering. From sizing toroidal tanks to estimating plasma volume in fusion reactors, the torus volume formula is both elegant and practically essential.
The Volume Formula
Where R is the major radius (center of torus to center of tube) and r is the minor radius (tube radius).
Pappus\'s Centroid Theorem
The torus volume formula is derived from Pappus\'s second centroid theorem: the volume of a solid of revolution equals the generating area (πr² for a circle) times the distance traveled by its centroid (2πR).
Worked Example
Calculate the volume of a torus with R = 5 cm and r = 2 cm:
- Calculate r²: 2² = 4
- Calculate 2π²: 2 × 9.8696 = 19.7392
- Multiply: 19.7392 × 5 × 4 = 394.78 cm³
- Result: The volume is approximately 394.78 cm³
Scaling Properties
Volume scales linearly with R and quadratically with r. Doubling R doubles the volume; doubling r quadruples it. This has important implications for engineering design.
Applications
- Tokamak reactors: Estimating plasma volume for fusion energy
- Manufacturing: Material volume for toroidal parts
- Food industry: Donut production volume calculations
- Architecture: Toroidal building and structure volumes