Torus Calculator
Calculate volume, surface area, and other properties of a torus (donut shape)
Calculation Results
Enter torus measurements to calculate all properties.
Major Radius (R): units
Minor Radius (r): units
Volume (V): cubic units
Surface Area (S): square units
Aspect Ratio:
Torus Diagram
Torus Formulas
Key Properties
- Major Radius (R): Distance from center of torus to center of tube
- Minor Radius (r): Radius of the circular tube
- Volume (V): Space contained within the torus
- Surface Area (S): Total area of the torus surface
- Aspect Ratio: Ratio of major to minor radius (R/r)
Calculation Formulas
- Volume (V): V = 2π²Rr²
- Surface Area (S): S = 4π²Rr
- Aspect Ratio: R/r
Where: R = major radius, r = minor radius, π ≈ 3.14159
Example Calculation
For a torus with:
Major radius (R) = 6 units, Minor radius (r) = 2 units.Calculations:
- Volume ≈ 2 × π² × 6 × 2² ≈ 473.74 units³
- Surface Area ≈ 4 × π² × 6 × 2 ≈ 473.74 units²
- Aspect Ratio = 6 / 2 = 3
About Toruses
A torus (plural: tori) is a doughnut-shaped surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a ring torus.
Real-World Applications
- Engineering: Toroidal transformers and inductors
- Physics: Tokamak fusion reactors use toroidal shapes
- Architecture: Some unique buildings use toroidal shapes
- Everyday Objects: Lifebuoys, donuts, and tire inner tubes
Special Cases
- Ring Torus: R > r (standard donut shape)
- Horn Torus: R = r (the hole just disappears)
- Spindle Torus: R < r (self-intersecting)
- Sphere: R = 0 (degenerate case)
Torus Components
Major Radius (R)
Distance from center to tube center
Minor Radius (r)
Radius of the tube
Volume (V)
V = 2π²Rr²
Surface Area (S)
S = 4π²Rr
How to Use the Torus Calculator
The Torus Calculator helps compute various properties based on input values. Here's how to use it:
1. Select Calculation Method
Choose what measurements you know:
- Major & Minor Radii - Enter both radius values
- Volume & Major Radius - To find the minor radius
- Surface Area & Major Radius - To find the minor radius
2. Enter Your Values
Input positive numbers in the appropriate fields:
Example 1: Major Radius = 6, Minor Radius = 2
Example 2: Volume = 473.74, Major Radius = 6
3. Get Results
The calculator will compute all properties:
- Major Radius (R)
- Minor Radius (r)
- Volume (V)
- Surface Area (S)
- Aspect Ratio (R/r)
Practical Applications
- Calculate material needed for toroidal structures
- Determine paint required for torus-shaped objects
- Find storage capacity of toroidal containers
- Solve physics problems involving toroidal shapes
Frequently Asked Questions
A sphere is perfectly symmetrical in all directions, while a torus has a hole through the center. Mathematically, a sphere can be considered a special case of a torus where the major radius is zero.
Yes, a flat torus is a theoretical concept in mathematics where the surface has zero curvature everywhere. It can't be physically constructed in 3D space without distortion, but it exists as an abstract concept.
Use the formula: r = √(V / (2π²R)) where V is volume and R is major radius.