A sphere is a two-dimensional surface embedded in three-dimensional space, while a ball is the interior of a sphere (which may or may not include the sphere itself).
A sphere and ball of radius r centered at the origin are respectively described by the equation and inequality,
x² + y² + z² = r²
and
x² + y² + z² < r²
or
x² + y² + z² ≤ r²
