GEOMETRY3D GeometryMathematics Calculator
🍩

Compute surface area and volume of a torus (donut shape)

Enter major radius R and minor radius r to get surface area (4π²Rr), volume (2π²Rr²), and inner/outer diameters.

Concept Fundamentals
A = 4π²Rr
Surface Area
V = 2π²Rr²
Volume
2(R − r)
Inner Diameter
2(R + r)
Outer Diameter

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Surface area A = 4π²Rr Volume V = 2π²Rr² R = major radius (center to tube center) r = minor radius (tube thickness) R must be > r for ring torus

Key quantities
A = 4π²Rr
Surface Area
Key relation
V = 2π²Rr²
Volume
Key relation
2(R − r)
Inner Diameter
Key relation
2(R + r)
Outer Diameter
Key relation

Ready to run the numbers?

Why: Tori appear everywhere—donuts, O-rings, life preservers. Surface area determines material needed; volume drives capacity calculations.

How: Pappus's centroid theorem: the torus is generated by rotating a circle. Surface area = circumference × path length = 2πr × 2πR = 4π²Rr.

Surface area A = 4π²RrVolume V = 2π²Rr²
Sources:Wolfram MathWorld — TorusKhan Academy — Solids of Revolution

Run the calculator when you are ready.

Calculate Torus PropertiesEnter R and r to get surface area and volume
🍩
3D GEOMETRYDonut Shape

Torus Surface Area

A = 4π²Rr. R = major radius, r = minor radius. Donuts, O-rings, life preservers.

🍩 Examples — Click to Load

Torus Dimensions

torussa_calc.sh
CALCULATED
$ calculate_torus_sa --R=4 --r=2
Surface Area
315.8273
Volume
315.8273
Inner Diam.
4
Outer Diam.
12
Share:
Torus Properties
R = 4, r = 2
315.83 cm²

3D Visualization

R = 4r = 2
Torus Visualization

Property Radar

Property Comparison

Property Breakdown

📐 Calculation Breakdown

INPUT
Major Radius (R)
4
Minor Radius (r)
2
RESULTS
Surface Area
315.8273 cm²
A = 4\text{pi} ^{2} ext{Rr}
Volume
315.8273 cm³
V = 2\text{pi} ^{2} ext{Rr}^{2}
Inner Diameter
4 cm
2(R - r)
Outer Diameter
12 cm
2(R + r)

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

🍩

Donut surface area = 4π²Rr — determines glaze coverage.

— Food

⭕

O-ring seals use inner/outer diameter for groove fit.

— Engineering

📐

Pappus's theorem: surface = generating curve × path length.

— Geometry

🛢️

Tire inner tube volume = 2π²Rr² for air capacity.

— Automotive

📋 Key Takeaways

  • • Torus surface area: A=4π2RrA = 4\pi^2 Rr — product of major and minor radii
  • • Volume: V=2π2Rr2V = 2\pi^2 Rr^2 — minor radius squared
  • • Inner diameter = 2(R − r), Outer diameter = 2(R + r)
  • • Pappus's theorem: A = (2πR)(2πr) — circumference × tube circumference

💡 Did You Know?

🍩Donuts, bagels, and life preservers are real-world toriSource: Everyday objects
⚛️Tokamak fusion reactors use toroidal chambers to contain plasmaSource: Nuclear fusion
📐Pappus of Alexandria (c. 300 CE) derived the torus formulasSource: History of math
🛞Bicycle inner tubes are tori — R is wheel radius, r is tube thicknessSource: Engineering
🔬Carbon nanotori have unique electronic propertiesSource: Nanotechnology
🎮Torus topology: genus 1 — one "hole" like a coffee cupSource: Topology
⭕O-rings are tori — surface area determines coating needsSource: Manufacturing

📖 How It Works

R = distance from center of torus to center of tube. r = tube radius.

Pappus's Second Theorem

Surface area = (length of generating curve) × (distance traveled by centroid). For a torus: curve = circle of circumference 2πr, centroid travels 2πR → A = 4π²Rr.

🎯 Expert Tips

💡 Ring vs Horn

Ring torus: R > r (donut). Horn: R = r. Spindle: R < r (self-intersecting).

💡 Doubling r

Doubling r doubles SA but quadruples volume (r² in volume formula).

💡 Unit Consistency

Use consistent length units. SA in unit², volume in unit³.

💡 O-Ring Sizing

For seals, inner diameter = 2(R−r) and outer = 2(R+r) define the groove dimensions.

⚖️ Comparison

PropertyFormula
Surface Area4π²Rr
Volume2π²Rr²
Inner Diameter2(R − r)
Outer Diameter2(R + r)

❓ FAQ

What is R vs r?

R = major radius (center of torus to center of tube). r = minor radius (tube thickness).

Why must R > r?

For a ring torus (donut shape). When R ≤ r, the shape self-intersects.

How is Pappus used?

Unroll the torus mentally: you get a cylinder with height 2πR and circumference 2πr. Lateral area = 4π²Rr.

What units?

Use consistent length units. Surface area in unit², volume in unit³.

How do I measure a donut?

R = distance from center of hole to center of the tube. r = half the tube thickness (minor radius).

What about tire inner tubes?

R ≈ wheel radius, r = tube cross-section radius. SA determines rubber needed.

Can I use this for O-rings?

Yes. Surface area helps estimate coating or seal material. Inner/outer diameter for groove fit.

📊 Stats

4π²
SA factor
2π²
Vol factor
1
Genus (hole)
0
Euler χ

⚠️ Disclaimer: For ring torus only (R > r). Results are mathematically precise. Verify for critical applications.

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