🍩 Torus Surface Area Calculator
Calculate the surface area of a torus (donut shape)
Distance from center to tube center
Radius of the tube
How to Use This Calculator
Enter Two Radii
Input the major radius (R) - distance from center of torus to center of the tube, and minor radius (r) - radius of the tube itself. Make sure R > r.
Click Calculate
Press the "Calculate Surface Area" button to compute the total surface area of the torus using the formula A = 4π²Rr.
Review Result
View the calculated surface area displayed in square units. The result includes the step-by-step calculation showing how the formula was applied.
Formula
Surface Area = 4π²Rr
Where R = major radius, r = minor radius
Where:
- R = major radius (distance from center of torus to center of tube)
- r = minor radius (radius of the circular tube)
- Surface Area = total area covering the torus surface
- π ≈ 3.14159
- Requirement: R > r (major radius must be greater than minor radius)
Note: A torus is a donut-shaped surface formed by rotating a circle around an axis. The formula comes from integrating the surface area of the rotated circle.
Example 1: Torus with major radius R = 5 units, minor radius r = 2 units
Surface Area = 4 × π² × 5 × 2
Surface Area = 4 × 9.8696 × 10
Surface Area ≈ 394.78 units²
Example 2: Torus with major radius R = 10 units, minor radius r = 3 units
Surface Area = 4 × π² × 10 × 3
Surface Area = 4 × 9.8696 × 30
Surface Area ≈ 1,184.35 units²
Example 3: Torus with major radius R = 8 units, minor radius r = 1.5 units
Surface Area = 4 × π² × 8 × 1.5
Surface Area ≈ 473.74 units²
About Torus Surface Area Calculator
A torus is a donut-shaped 3D surface formed by rotating a circle around an axis in 3D space. It has two radii: the major radius (R) - distance from the center of the torus to the center of the tube, and the minor radius (r) - the radius of the circular tube itself. This calculator finds the surface area using the formula A = 4π²Rr.
When to Use This Calculator
- Engineering: Calculate surface area for torus-shaped components or structures
- Architecture: Design torus-shaped buildings or decorative elements
- Manufacturing: Determine material requirements for torus-shaped products
- Mathematics Education: Teach students about surfaces of revolution and torus geometry
- 3D Modeling: Work with torus shapes in CAD or graphics software
- Physics: Calculate surface area for torus-shaped objects in physics problems
Why Use Our Calculator?
- ✅ Simple Formula: Just enter two radii to get surface area
- ✅ Instant Results: Get accurate calculations immediately
- ✅ Step-by-Step Display: See the formula applied with your values
- ✅ 100% Accurate: Uses precise mathematical formula
- ✅ Educational: Helps understand torus geometry
- ✅ Completely Free: No registration required
Understanding Torus Geometry
A torus is created by rotating a circle:
- Major Radius (R): Distance from the center of the torus to the center of the circular tube
- Minor Radius (r): Radius of the circular tube itself
- Requirement: R > r (the tube must fit within the torus)
- Surface Area: Total area covering the entire torus surface
- Shape: Resembles a donut or inner tube
Real-World Applications
Engineering: A torus-shaped pipe fitting has major radius 15 cm and minor radius 2 cm. Surface area = 4π² × 15 × 2 ≈ 1,184.35 cm². This determines material needed for manufacturing.
Architecture: A torus-shaped structure with major radius 20 m and minor radius 3 m has surface area = 4π² × 20 × 3 ≈ 2,368.69 m² (coating or finishing material needed).
Manufacturing: A torus component with R = 8 cm and r = 1.5 cm has surface area ≈ 473.74 cm² (material required for production).
Frequently Asked Questions
What is a torus?
A torus is a donut-shaped 3D surface formed by rotating a circle around an axis in 3D space. It has a hole in the middle and resembles a donut or inner tube. It's defined by two radii: major radius (R) and minor radius (r).
What's the difference between major and minor radius?
Major radius (R) is the distance from the center of the torus to the center of the circular tube. Minor radius (r) is the radius of the circular tube itself. The major radius must be greater than the minor radius (R > r).
Why is the formula 4π²Rr?
This formula comes from calculating the surface area of a torus using calculus (surface of revolution). When you rotate a circle of radius r around an axis at distance R, the resulting surface area is 4π²Rr. The π² appears from integrating the rotation.
What happens if R = r?
If R = r, the torus becomes a degenerate case (the tube would touch the center). For a valid torus, R must be greater than r. If they're equal, the formula would still work mathematically, but it wouldn't represent a typical torus shape.
Can I calculate volume too?
Yes! Torus volume = 2π²Rr². This is different from surface area. Volume measures the space inside the solid torus, while surface area measures the area covering the surface. Use a separate torus volume calculator for volume calculations.
What are some real-world examples of toruses?
Donuts, inner tubes, life preservers, bagels, and some types of pipe fittings are torus-shaped. In mathematics and physics, toruses appear in topology, magnetic field configurations, and particle accelerator designs.