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Torus Surface Area Calculator
Calculate torus (donut) surface area, volume, and geometric properties
📚 Examples, Rules & Help
⚡Quick Examples of Torus Surface Area
🔍How to Calculate Torus Surface Area
🍩 Understanding Torus Surface Area
A torus (donut shape) is formed by rotating a circle around an external axis. It's characterized by two radii: the major radius (R) from the center to the tube center, and the minor radius (r) of the tube itself.
Essential Formulas:
Surface Area: SA = 4π²Rr
Volume: V = 2π²Rr²
Constraint: r < R (minor radius must be less than major radius)
Outer Diameter: D_outer = 2(R + r)
Inner Diameter: D_inner = 2(R - r)
🌍Real-World Applications
🍩 ⚙️ Engineering & Design
Mechanical and architectural applications
❓Frequently Asked Questions
What's the difference between major and minor radius?
Major Radius (R): Distance from the torus center to the center of the tube
Minor Radius (r): Radius of the circular tube itself
Think of a donut: R is how far the donut hole center is from the tube center, r is how thick the donut tube is.
Why must the minor radius be smaller than the major radius?
If r ≥ R, the shape would not be a proper torus:
If r = R: The inner hole would close completely
If r > R: The shape would intersect itself
A valid torus always has an open center hole, requiring r < R.
🎯Common Use Cases
⚙️ Engineering & Manufacturing
- O-ring and gasket design
- Tire and tube calculations
- Donut-shaped components
- Ring gear surface area
🏗️ Architecture & Design
- Circular building elements
- Sculptural torus shapes
- Ring-shaped structures
- Decorative donut elements
💡Calculator Tips & Best Practices
📏Radius Constraint
Always ensure minor radius (r) < major radius (R) for a valid torus shape.
💡Double Pi Squared
Torus formulas uniquely involve 4π² and 2π² due to the double revolution geometry.
📚 References & Further Reading
Engineering principles for torus-shaped components and their optimization in various industries
External Link
Note: These references provide additional mathematical context and verification of the formulas used in this calculator.