GEOMETRY3D GeometryMathematics Calculator
🍩

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

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

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

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Volume V = 2π²Rr² Surface area A = 4π²Rr Volume ∝ r² — doubling r quadruples volume R = major radius, r = minor radius V/A ratio = r/2

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

Ready to run the numbers?

Why: Torus volume drives donut dough amount, O-ring capacity, and tire tube air volume. Essential for manufacturing and packaging.

How: The volume comes from Pappus's centroid theorem: the torus is generated by rotating a circle of area πr². Volume = 2πR × πr² = 2π²Rr².

Volume V = 2π²Rr²Surface area A = 4π²Rr
Sources:Wolfram MathWorld — TorusKhan Academy — Solids of Revolution

Run the calculator when you are ready.

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

Torus Volume

V = 2π²Rr². R = major radius, r = minor radius.

🍩 Examples — Click to Load

Torus Dimensions

torusvol_calc.sh
CALCULATED
$ calculate_torus_vol --R=4 --r=2
Volume
315.8273
Surface Area
315.8273
Inner Diam.
4
Outer Diam.
12
Share:
Torus Volume
R = 4, r = 2
315.83 cm³

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
Volume
315.8273 cm³
V = 2\text{pi} ^{2} ext{Rr}^{2}
Surface Area
315.8273 cm²
A = 4\text{pi} ^{2} ext{Rr}
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 volume = 2π²Rr² — determines dough needed.

— Food

⭕

O-ring volume helps estimate seal material.

— Engineering

📐

Pappus's theorem: volume = generating area × path length.

— Geometry

🛢️

Doubling r quadruples volume (V ∝ r²).

— Scaling

📋 Key Takeaways

  • • Torus volume: V=2π2Rr2V = 2\pi^2 Rr^2 — minor radius squared
  • • Surface area: A=4π2RrA = 4\pi^2 Rr
  • • V/A ratio = r/2 — depends only on minor radius
  • • Pappus: V = (area of circle πr²) × (path 2πR)

💡 Did You Know?

🍩Donut volume determines how much dough or filling is neededSource: Baking
⚛️ITER tokamak has 840 m³ plasma volume in its toroidal chamberSource: Fusion
📐Doubling r quadruples volume (r²); doubling R doubles volumeSource: Scaling
🛞Inner tube volume = air capacity for inflationSource: Engineering
🔬Torus has genus 1 — topologically like a coffee cupSource: Topology
⭕O-ring volume determines seal material costSource: Manufacturing

📖 How It Works

Pappus's First Theorem: Volume of solid of revolution = (area of generating shape) × (distance traveled by centroid).

Generating circle area = πr². Centroid travels 2πR. So V = πr² × 2πR = 2π²Rr².

🎯 Expert Tips

💡 Minor Radius Effect

Volume ∝ r². Small change in r has big impact on volume.

💡 Fluid Capacity

Volume in cm³ ÷ 1000 = liters. Useful for tanks and tubes.

💡 Ring vs Horn

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

💡 V/A Ratio

V/A = r/2 — depends only on minor radius. Thicker tubes have higher V/A.

⚖️ Comparison

PropertyFormula
Volume2π²Rr²
Surface Area4π²Rr
V/A Ratior/2

❓ FAQ

Why is volume proportional to r²?

The generating circle has area πr². As it sweeps, each "slice" contributes. The r² comes from the circular cross-section.

How to convert to liters?

Volume in cm³ ÷ 1000 = liters. For in³, ÷ 61.024.

What if R = r?

Horn torus — volume formula still applies but shape self-intersects at one point.

What is R vs r?

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

How much dough for a donut?

Volume gives the dough needed. Multiply by density for weight. Typical donut ~50–100 cm³.

Inner tube air capacity?

Volume in cm³ ÷ 1000 = liters of air at atmospheric pressure when inflated.

Why does doubling r quadruple volume?

Volume has r². (2r)² = 4r², so volume quadruples.

📊 Stats

2π²
Vol factor
4π²
SA factor
r/2
V/A ratio
1
Genus

⚠️ Disclaimer: For ring torus (R > r). Results are mathematically precise.

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