Torus Calculator | Volume & Surface Area
Calculate the volume, surface area, and other properties of a torus (donut shape).
Quick Examples
Torus Geometry
R = major radius (centre to tube centre)
r = minor radius (tube radius)
Require r < R for a valid torus
What Is the Torus Calculator | Volume & Surface Area?
A torus is the 3D surface generated by revolving a circle of radius r around an axis in the same plane, at a distance R from the circle's centre. Familiar examples include donuts, O-rings, inflatable tubes, and tire cross-sections.
Pappus's Centroid Theorem
- ›Volume = cross-section area × distance travelled by centroid = πr² × 2πR = 2π²Rr²
- ›Surface area = cross-section perimeter × distance travelled = 2πr × 2πR = 4π²Rr
- ›This elegant theorem applies to any solid of revolution, not just the torus
- ›Requirement: r < R for a ring torus (the common donut shape)
Types of Torus
- ›Ring torus (r < R): hole in the centre, the common donut shape
- ›Horn torus (r = R): tube passes through itself at a single point
- ›Spindle torus (r > R): self-intersecting; more like a sphere with indentations
- ›This calculator computes ring torus geometry only (requires r < R)
Formula
Volume
V = 2π²Rr²
Surface Area
SA = 4π²Rr
Inner Radius
R_inner = R − r
Outer Radius
R_outer = R + r
Mean Circumference
C_mean = 2πR
Tube Circumference
C_tube = 2πr
How to Use
- 1Enter the Major Radius R, the distance from the torus centre to the centre of the tube.
- 2Enter the Minor Radius r, the radius of the tube itself.
- 3Ensure r < R for a valid ring torus (the calculator validates this).
- 4Select the unit of measurement (mm, cm, m, in, ft).
- 5Click Calculate, results show volume, surface area, SA:Volume ratio, and all geometry.
- 6Use the Quick Example presets to explore typical torus shapes (donut ring, O-ring, tire).
Example Calculation
Standard donut ring: R = 5 cm (major radius), r = 2 cm (tube radius):
Volume:
V = 2π² × 5 × 2²
= 2 × 9.8696 × 5 × 4
= 394.78 cm³
Surface Area:
SA = 4π² × 5 × 2
= 4 × 9.8696 × 10
= 394.78 cm²
Inner radius = 5 − 2 = 3 cm
Outer radius = 5 + 2 = 7 cm
Interesting Coincidence
For R = 5 and r = 2, volume (cm³) numerically equals surface area (cm²). This occurs when SA/V = 1, i.e. when r = R/2. The SA:Volume ratio simplifies to 2/r, independent of R entirely.
Understanding Torus | Volume & Surface Area
Torus Geometry, Key Properties Summary
| Property | Formula | Example (R=5, r=2 cm) | Notes |
|---|---|---|---|
| Volume | V = 2π²Rr² | 394.78 cm³ | Pappus's theorem |
| Surface Area | SA = 4π²Rr | 394.78 cm² | Pappus's theorem |
| SA:Volume ratio | 2/r | 1.000 cm⁻¹ | Independent of R |
| Inner radius | R − r | 3 cm | Hole's inner edge |
| Outer radius | R + r | 7 cm | Full extent |
| Outer diameter | 2(R + r) | 14 cm | Bounding circle |
| Inner circumference | 2π(R − r) | 18.85 cm | Inner ring |
| Outer circumference | 2π(R + r) | 43.98 cm | Outer ring |
| Tube circumference | 2πr | 12.57 cm | Cross-section perimeter |
| Mean path length | 2πR | 31.42 cm | Centreline of tube |
Frequently Asked Questions
What is a torus in geometry?
A torus is a surface of revolution, a circle rotated around an axis in its plane:
- ›R (major radius): distance from the torus centre (hole centre) to the centre of the tube
- ›r (minor radius): radius of the tube cross-section itself
- ›Familiar shapes: donuts, O-rings, inflatable pool rings, Tokamak fusion chambers
- ›Mathematical property: a torus is topologically equivalent to a coffee mug (both have one hole)
What is Pappus's Centroid Theorem?
- ›Volume = (area of cross-section) × (distance centroid travels) = πr² × 2πR = 2π²Rr²
- ›Surface area = (perimeter of cross-section) × (distance centroid travels) = 2πr × 2πR = 4π²Rr
- ›The centroid of a circle is its centre, which travels a path of circumference 2πR
- ›This theorem was proved by Pappus of Alexandria around 320 AD, one of the oldest results in calculus-adjacent mathematics
What is the difference between major and minor radius?
- ›R = major radius: measures how "wide" the torus is overall (centre to tube centre)
- ›r = minor radius: measures how "thick" the tube is (half the tube diameter)
- ›Outer radius of the torus = R + r; inner radius (hole's edge) = R − r
- ›For a valid ring torus with a visible hole, we need r < R; if r = R the hole disappears
What real objects are shaped like a torus?
- ›O-rings: rubber seals for pipes and hydraulic fittings, precise torus geometry is critical for sealing
- ›Tire cross-sections: the tyre tread area wraps around a toroidal form
- ›Tokamak fusion reactors: the plasma is confined in a toroidal magnetic field chamber
- ›Toroidal inductors: wire wound on a donut-shaped ferromagnetic core, used in power supplies
What happens when r equals R?
- ›r < R: ring torus, donut with a visible hole. This calculator handles this case
- ›r = R: horn torus, the inner hole shrinks to a point; the tube just touches the axis
- ›r > R: spindle torus, the tube self-intersects; generates a shape closer to a sphere
- ›At r = R the inner radius (R − r) = 0, meaning the inner edge degenerates to a single point
How do I find the volume of an O-ring?
- ›Measure the O-ring cross-section diameter (the thickness of the rubber) → r = diameter/2
- ›Measure the inner diameter of the O-ring (the hole) → inner radius = R − r → solve for R
- ›Example: cross-section = 4 mm (r = 2 mm), inner diameter = 20 mm (inner radius = 10 mm) → R = 12 mm
- ›V = 2π² × 12 × 2² = 947 mm³ of rubber, useful for material cost estimation
How does the SA:Volume ratio change with torus size?
- ›SA/V = (4π²Rr) / (2π²Rr²) = 2/r, the R cancels out completely
- ›Only the tube radius r matters: thinner tubes (smaller r) have higher SA:V
- ›High SA:V is desirable in heat exchangers, more surface per unit of fluid volume
- ›Example: r = 1 mm gives SA/V = 2,000 m⁻¹; r = 10 mm gives 200 m⁻¹, 10× difference
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