Volume of a Torus
Calculate the volume and surface area of a torus (doughnut shape) using the radii formula.
Step-by-step examples included.
The Formulas
Surface Area: A = 4π²Rr
A torus is a three-dimensional shape formed by revolving a circle around an axis that lies in the same plane as the circle but does not intersect it. The most familiar example is a doughnut shape.
The torus is defined by two radii. The major radius (R) is the distance from the center of the torus to the center of the tube. The minor radius (r) is the radius of the tube itself. These two measurements completely determine the size and shape of the torus.
The volume formula can be understood through Pappus's centroid theorem: the volume of a solid of revolution equals the area of the cross-section (πr²) multiplied by the distance traveled by its centroid (2πR). This gives V = πr² × 2πR = 2π²Rr². Similarly, the surface area equals the circumference of the cross-section (2πr) multiplied by the centroid path (2πR), giving A = 4π²Rr.
Variables
| Symbol | Meaning |
|---|---|
| V | Volume of the torus |
| A | Surface area of the torus |
| R | Major radius — center of torus to center of tube |
| r | Minor radius — radius of the tube |
| π | Pi ≈ 3.14159 |
Example 1
A doughnut has a major radius of 8 cm and a tube radius of 3 cm. What is its volume?
Identify values: R = 8 cm, r = 3 cm
V = 2π²Rr² = 2 × (3.14159)² × 8 × 3²
V = 2 × 9.8696 × 8 × 9 = 2 × 9.8696 × 72
V = 2 × 710.61
V ≈ 1,421.2 cm³
Example 2
An inflatable tube (pool ring) has R = 40 cm and r = 10 cm. What is its surface area?
Identify values: R = 40 cm, r = 10 cm
A = 4π²Rr = 4 × (3.14159)² × 40 × 10
A = 4 × 9.8696 × 400 = 4 × 3,947.84
A ≈ 15,791.4 cm² (about 1.58 m²)
When to Use It
Torus calculations come up in engineering, manufacturing, and design.
- Calculating the volume of O-rings and gaskets
- Designing toroidal transformers and inductors in electronics
- Computing material needed for inflatable rings and tubes
- Architecture and product design with ring-shaped elements