Detailed definition
Understanding Torus
A torus is formed by rotating a circle around an axis that lies in the same plane as the circle but does not cut through it. The result is the familiar ring or doughnut-shaped surface.
Unlike the sphere, the torus has a central hole, which changes many of its geometric and topological properties. That makes it a useful contrast object in higher-level geometry and topology.
This page keeps the generating circle idea tied to the finished ring shape so the torus is understood as a surface of revolution rather than as a strange isolated solid name.
Key facts
Important ideas to remember
- A torus is a doughnut-shaped surface formed by rotating a circle around an axis outside the circle.
- A torus can be viewed as a surface of revolution.
- Its shape depends on both the radius of the generating circle and the distance from that circle to the axis of rotation.
- The torus is not simply a bent cylinder; its closed ring structure gives it a very different geometry.
Where it is used
Where torus shows up
- Use the torus in advanced geometry, topology, and modelling discussions.
- Use it in contexts involving ring-shaped solids or surfaces, such as seals, tubes, and reactors.
- Use it when comparing surfaces of revolution with more familiar solids like spheres and cones.
Common mistakes
What to watch out for
- Do not treat a torus as if it were just a sphere with a dent; the central hole is structurally essential.
- Do not ignore the rotating-circle construction that defines the shape.
- Do not confuse a torus with a cylinder or an annulus, which are different geometric objects.