Definition of a sphere

Question

If there is no further specification (such as solid or hollow), does a "sphere" refer to the solid/filled form or the hollow shell?

Thanks.

Answer 1

In mathematics, the word "sphere" refers to the hollow shell (see here). The word "ball" is reserved for the solid version (see here).

So, for example, the unit sphere in $\mathbb{R}^3$ is equal to the set $$\{(x,y,z)\in\mathbb{R}^3\mid x^2+y^2+z^2=1\}$$ while the unit ball in $\mathbb{R}^3$ is equal to the set $$\{(x,y,z)\in\mathbb{R}^3\mid x^2+y^2+z^2\leq 1\}.$$

In common usage of the word "sphere" in English, it could mean either.

Answer 2

From Wikipedia:

A sphere [...] is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius of the sphere. The maximum straight distance through the sphere is known as the diameter of the sphere. It passes through the center and is thus twice the radius.

In higher mathematics, a careful distinction is made between the sphere (a two-dimensional spherical surface embedded in three-dimensional Euclidean space) and the ball (the three-dimensional shape consisting of a sphere and its interior).

That is, if you're speaking in everyday terms, it's ambiguous; but if you're speaking to trained mathematicians (as opposed even to mathematical scientists, such as physicists), they will probably understand it to mean the surface at a constant distance from a central point — not only in three dimensional space, but in any Euclidean space.