A torus is the donut-shaped solid formed when a circle is rotated around an axis outside the circle. Its geometry appears in tires, lifebuoys, magnetic confinement devices, and many design problems involving circular symmetry. To describe a standard torus, we use two radii: the major radius R from the center of the hole to the center of the tube, and the minor radius r of the circular tube.
Knowing these two values lets us calculate both volume and surface area exactly.
The volume formula comes from imagining the circular cross section traveling around a circular path. By Pappus's centroid theorem, the volume equals the area of the rotating circle times the distance traveled by its centroid. Since the circle has area pi r^2 and its centroid travels a distance 2 pi R, the volume is V = 2 pi^2 R r^2.
The surface area follows the same idea: the circumference of the tube circle, 2 pi r, travels around the same path 2 pi R, giving A = 4 pi^2 R r.
Key Facts
- Major radius R is the distance from the center of the torus to the center of the tube.
- Minor radius r is the radius of the circular tube.
- Volume of a torus: V = 2 pi^2 R r^2.
- Surface area of a torus: A = 4 pi^2 R r.
- Pappus volume theorem: volume = area of generating shape times distance traveled by its centroid.
- For a standard ring torus, the axis of rotation must not cut through the circular cross section, so R > r.
Vocabulary
- Torus
- A torus is a donut-shaped surface or solid formed by rotating a circle around an external axis in the same plane.
- Major radius
- The major radius R is the distance from the center of the torus to the center of the circular tube.
- Minor radius
- The minor radius r is the radius of the circular cross section that makes up the tube.
- Centroid
- The centroid is the geometric center or balance point of a shape.
- Pappus's centroid theorem
- Pappus's centroid theorem relates a volume or surface of revolution to the distance traveled by the centroid of the generating shape or curve.
Common Mistakes to Avoid
- Using the outside radius as R without checking the diagram is wrong because R measures to the center of the tube, not to the outer edge.
- Forgetting to square r in V = 2 pi^2 R r^2 is wrong because the circular cross section area is pi r^2.
- Swapping the volume and surface area formulas is wrong because volume uses an area times a distance, while surface area uses a length times a distance.
- Applying the standard ring torus formulas when R is less than or equal to r can be wrong because the shape becomes a horn or spindle torus rather than a simple ring torus.
Practice Questions
- 1 A torus has major radius R = 6 cm and minor radius r = 2 cm. Find its volume in terms of pi and approximate it using pi = 3.14.
- 2 A torus has R = 10 m and r = 1.5 m. Find its surface area in square meters using pi = 3.14.
- 3 Explain why Pappus's theorem uses the path traveled by the centroid of the circular cross section, not the path traveled by the outside edge of the torus.