What Is a Torus?

How the hole and tube arise

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.

You can picture a torus by moving a small circle around a larger circular path without letting it pass through the path's center. The small circle sweeps out the tube, while the empty middle remains a real part of the shape. That construction explains the hole more clearly than a front-view sketch.

Changing the radius of the tube changes the thickness of the torus, while changing the distance from the axis changes the size of the central opening. The two measurements describe different circular motions in the construction.

• 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 tori help

• 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.

Torus checks to make

• 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.