Torus / Donut shape (3D) — free printable diagram

What a torus is, and how to count it

A torus is a 3D solid that has no flat faces, 0 edges and 0 vertices. Part of its surface is curved, so it is not a polyhedron and it is counted differently from a cube or a prism — which is also why Euler's formula does not apply to it.

Solid

Torus (donut shape)

Surfaces

1 curved surface, no flat faces

Flat faces

0

Edges

0

Vertices (corners)

0

Is it a polyhedron?

No — it has a curved surface

Volume

V = 2π²Rr², where R is the distance from the centre of the hole to the centre of the tube and r is the radius of the tube

Surface area

SA = 4π²Rr

• A torus is the surface swept out when a circle is rotated all the way round a line that does not cut through it — which is why it takes two radii to describe, one for the tube and one for the ring.
• It has no edges and no vertices, like a sphere, but it is not the same kind of surface: it has a hole through it. For a torus, faces + vertices − edges comes to 0 rather than 2, so even the generalised form of Euler's formula gives a different answer. That hole is the reason.
• A doughnut, a bagel and an inner tube are all toruses (or tori).