🍩 Torus Volume Calculator
Calculate the volume 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 Volume" button to compute the volume of the solid torus using the formula V = 2π²Rr².
Review Result
View the calculated volume displayed in cubic units. The result includes the step-by-step calculation showing how the formula was applied.
Formula
Volume = 2π²Rr²
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)
- Volume = space inside the solid torus
- π ≈ 3.14159
- Requirement: R > r (major radius must be greater than minor radius)
Note: This formula gives the volume of the solid torus (filled donut). The formula comes from the Pappus centroid theorem or by integrating the volume of revolution.
Example 1: Torus with major radius R = 5 units, minor radius r = 2 units
Volume = 2 × π² × 5 × 2²
Volume = 2 × 9.8696 × 5 × 4
Volume ≈ 394.78 units³
Example 2: Torus with major radius R = 10 units, minor radius r = 3 units
Volume = 2 × π² × 10 × 3²
Volume = 2 × 9.8696 × 10 × 9
Volume ≈ 1,776.53 units³
Example 3: Torus with major radius R = 8 units, minor radius r = 1.5 units
Volume = 2 × π² × 8 × 1.5²
Volume = 2 × 9.8696 × 8 × 2.25
Volume ≈ 355.31 units³
About Torus Volume Calculator
A torus is a donut-shaped 3D solid 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 volume of the solid torus using the formula V = 2π²Rr².
When to Use This Calculator
- Engineering: Calculate volumes for torus-shaped tanks, pipes, or components
- Manufacturing: Determine material volumes for torus-shaped products
- Architecture: Calculate volumes for torus-shaped structures or spaces
- Mathematics Education: Teach students about volumes of revolution and torus geometry
- 3D Modeling: Work with torus volumes in CAD or graphics software
- Physics: Calculate volumes for torus-shaped objects in physics problems
Why Use Our Calculator?
- ✅ Simple Formula: Just enter two radii to get volume
- ✅ 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 and volumes of revolution
- ✅ Completely Free: No registration required
Understanding Torus Volume
The torus volume formula comes from:
- Pappus Centroid Theorem: Volume = (Area of generating shape) × (Distance traveled by centroid) = (πr²) × (2πR) = 2π²Rr²
- Major Radius (R): Distance from center of torus to center of the circular tube
- Minor Radius (r): Radius of the circular tube itself
- Requirement: R > r (the tube must fit within the torus)
- Units: Volume is in cubic units (m³, cm³, etc.)
Real-World Applications
Engineering: A torus-shaped storage tank has major radius 15 m and minor radius 2 m. Volume = 2π² × 15 × 4 ≈ 1,184.35 m³. This determines the storage capacity.
Manufacturing: A torus-shaped component with R = 8 cm and r = 1.5 cm has volume = 2π² × 8 × 2.25 ≈ 355.31 cm³ (material volume needed for production).
Architecture: A torus-shaped building element with major radius 20 m and minor radius 3 m has volume = 2π² × 20 × 9 ≈ 3,553.11 m³ (space inside the structure).
Frequently Asked Questions
What is the difference between torus volume and surface area?
Volume (V = 2π²Rr²) measures the space inside the solid torus (cubic units). Surface area (A = 4π²Rr) measures the area covering the surface (square units). Volume is larger than surface area would suggest because it includes the entire 3D space inside.
Why is the formula 2π²Rr²?
This formula comes from the Pappus centroid theorem. When you rotate a circle of area πr² around an axis at distance R, the volume of revolution is (area) × (circumference of path) = (πr²) × (2πR) = 2π²Rr².
Can I calculate if I only know the surface area?
If you know surface area A = 4π²Rr, you have one equation with two unknowns (R and r). You'd need additional information (like the ratio R/r or one of the radii) to determine volume uniquely. The calculator requires both R and r.
What happens if R = r?
If R = r, it becomes a degenerate case where the tube touches the center. For a valid torus, R must be greater than r. Mathematically, if R = r, volume = 2π²R³, but this doesn't represent a typical torus shape.
How does this relate to a cylinder?
A torus is formed by rotating a circle, while a cylinder is formed by translating a circle. The formulas are different: cylinder volume = πr²h (where h is height), while torus volume = 2π²Rr² (where R is the rotation radius).
What are practical applications of torus volumes?
Torus volumes are used in: storage tank design (toroidal tanks), pipe fittings and joints, architectural elements, donut-shaped structures in engineering, magnetic field calculations in physics, and computational geometry applications.