Torus Calculator
Calculate the volume, surface area, and other properties of a torus (donut shape) using its major and minor radii.
Torus Volume and Surface Area Calculator
Calculate the volume and surface area of a torus (donut shape) with our easy-to-use calculator. Learn about torus formulas, applications, and explore interactive examples.
A torus is a three-dimensional shape that looks like a donut. It's formed by rotating a circle around an axis that lies in the same plane as the circle. This versatile shape appears in architecture, engineering, and even in atomic and molecular structures.
Formula
Volume = 2π²Rr²
Surface Area = 4π²Rr
Where:
- R:major radius (distance from center of tube to center of torus)
- r:minor radius (radius of the tube)
How to Calculate Torus Volume
The volume formula involves both the major radius (R) and minor radius (r). The factor 2π² comes from the double rotation: the circle being rotated (2πr) and the distance it travels (2πR).
Understanding Surface Area
The surface area formula is derived from the circumference of the generating circle (2πr) multiplied by the circumference of its rotational path (2πR). This gives us 4π²Rr.
Real-World Applications
Tori (plural of torus) are found in architecture (curved arches), engineering (rotating machinery), physics (magnetic fields), and even in everyday objects like inner tubes and donuts.
Frequently Asked Questions
What's the difference between major and minor radius?
The major radius (R) is the distance from the center of the torus to the center of the tube, while the minor radius (r) is the radius of the tube itself. Think of R as the "donut's radius" and r as the "thickness of the donut".
Can a torus have different major and minor radii?
Yes! When R > r, you get a ring torus (like a donut). When R = r, you get a horn torus. When R < r, you get a self-intersecting spindle torus. Each type has unique properties and applications.
Where are tori used in the real world?
Tori appear in magnetic fields around planets, smoke rings, architectural features (curved arches), plumbing fixtures (O-rings), and even in particle accelerators. They're also studied in quantum mechanics and topology.