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.
Understanding Geometry: Volume of a Torus
The two radii affect a torus in very different ways. Increasing the major radius makes the whole shape wider while leaving the tube thickness unchanged. Its volume therefore increases in direct proportion to the major radius.
Increasing the minor radius thickens the tube. This has a much stronger effect because the cross section is an area, and area depends on the radius squared.
If the tube radius doubles, the torus volume becomes four times as large when the major radius stays fixed. This square relationship is a useful check on any answer.
Units are especially important in volume problems. If both radii are measured in centimetres, the final volume must be in cubic centimetres. A length squared creates square units, then multiplying by another length creates cubic units.
The constant involving pi has no units. Students sometimes mix centimetres with metres in the same calculation, which can cause a large error.
Convert every measurement to one unit first. Keep extra digits for pi during the calculation, then round only at the end to a sensible number of decimal places.
A numerical estimate can make the formula less abstract. Suppose a torus has a major radius of ten centimetres and a tube radius of two centimetres. The tube cross section has an area of about twelve point six square centimetres.
The centre of that cross section travels about sixty two point eight centimetres around its circular route. Multiplying these quantities gives about seven hundred ninety cubic centimetres.
This method shows why the answer is reasonable. It is the area of one slice multiplied by the total distance that slice moves.
The familiar ring shape is only one member of a wider family. When the major radius equals the minor radius, the inner hole just closes at one point. This is called a horn torus.
When the major radius is smaller than the minor radius, the rotating circle overlaps itself, producing a self intersecting shape. The usual volume rule based on a traveling cross section no longer describes a simple solid in the same straightforward way. In school problems, check that the major radius is greater than the minor radius before applying the standard result.
In real objects such as inner tubes, seals, bagels, and curved pipes, the cross section may not be perfectly circular. The torus model is then an approximation, but it can still give a useful estimate when the dimensions are close to circular.
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.