Calculus How To

Closed Surface: Simple Definition, Examples

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A closed surface contains a volume of space, enclosed from all directions; It consists of one connected, hollow piece that has no holes and doesn’t intersect itself.

A closed surface has a natural positive direction, where a unit normal vector N points away from the interior (i.e. points toward the outside). A normal vector pointed outward indicates the “negative” direction.

closed surface vector

The two parts of this shape are oriented by two unit normal vectors.

Closed Surface Examples

examples of closed surface

Closed surfaces (L to R): Cube, Polyhedron, Sphere, Torus.

Examples of closed surfaces:

  • Cube,
  • Polyhedron,
  • Sphere,
  • Torus (an inflated inner tube).
  • Gaussian surface: any closed surface through which an electric field passes [1].

A Few examples of surfaces that are not closed: a plane, a sphere with a point removed, a crumpled tin can with a cross-section that looks like a figure-8 (it intersects itself), an infinite cylinder [2].

Do Closed Surfaces have a Boundary?

Möbius strip

A Möbius strip [3].

The interior of a closed surface is a completely enclosed, single finite region of space; The closed surface itself is the boundary of this region [2]. Sometimes closed surfaces are described as having no boundary [4], but there are cases where closed surfaces do have boundary (for example, a cylinder or Möbius strip). To avoid this problem with classification, many discussions on closed surfaces concentrate on simple closed surfaces, like cubes and spheres [5].

Calculating Integrals

When you have many surfaces (for example, a pyramid has four), calculating integrals becomes a challenge. The divergence theorem allows you to combine multiple surface integrals and use one triple (volume) integral instead.


[1] Chapter 24 Lecture. Retrieved April 22, 2021 from:
[2] MIT. (2007). 18.02 Multivariable Calculus. Retrieved April 22, 2021 from: CC Share Alike 4.0.
[3] Lntcnnmc, CC BY-SA 3.0 , via Wikimedia Commons
[4] V9. Surface Integrals. Retrieved April 22, 2021 from:
[5] Classification of Surfaces. Retrieved April 22, 2021 from:

Stephanie Glen. "Closed Surface: Simple Definition, Examples" From Calculus for the rest of us!

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