Torus — Volume + Surface Area
Torus (donut): V = 2π²·R·r², SA = 4π²·R·r. R = major (center to tube center), r = minor (tube radius).
Result
Volume
1,776.5288
2π²Rr² = 1,776.5288. SA = 4π²Rr = 1,184.3525.
- Major radius (R)10
- Minor radius (r)3
- Outer radius (R + r)13.000000
- Inner hole radius (R − r)7.000000
- Volume (2π²Rr²)1,776.528792
- Surface area (4π²Rr)1,184.352528
Step-by-step
- Pappus's theorem: V = 2πR × (πr²) = 2π²Rr² = 1,776.5288.
- Pappus's theorem: SA = 2πR × (2πr) = 4π²Rr = 1,184.3525.
- Outer = R + r = 13.0000; inner hole radius = R − r = 7.0000.
How to use this calculator
- Enter major radius R (axis-to-tube-center) and minor radius r (tube cross-section radius).
- Keep R ≥ r to avoid self-intersection.
- Read volume and surface area.
About this calculator
A torus is the surface formed by revolving a circle of radius r around an axis at distance R from the circle's center, where R ≥ r (otherwise the torus self-intersects). By Pappus's centroid theorem (~300 AD): V = 2πR × area-of-cross-section = 2πR × πr² = 2π²Rr², and SA = 2πR × perimeter-of-cross-section = 2πR × 2πr = 4π²Rr. Source: Apostol, "Calculus" Vol II Ch. 14; Pappus of Alexandria. The donut, inner tube, and bagel are tori; particle accelerators (Tevatron, LHC) approximate tori on a much larger scale.
Frequently asked
For a surface or volume of revolution: SA = path × perimeter, V = path × area. Path = 2πR (centroid travels a circle of radius R). Cross-section = circle of radius r. Yields 4π²Rr and 2π²Rr² immediately.
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