Maths calls the shell a sphere and the solid a ball
A sphere is the set of points at equal distance r from a centre in three-dimensional space—the surface analogue of a circle. Radius names both the segment and its length; a diameter through the centre is the longest chord. Mathematicians distinguish that hollow sphere from a solid ball.
The word comes from Greek sphaîra, ball, and the earliest known discussions appear in Greek mathematics. A diameter, a radius continued through the centre, is twice the radius and the longest segment joining two points on the surface, and its endpoints are called antipodal. Geometers treat the sphere as a two-dimensional closed surface sitting in three-dimensional space and keep it separate from the ball it encloses: an open ball leaves the surface out, and a closed ball includes it.
On the surface, the axis through the centre meets it at two poles, the equator is the great circle equidistant from them, meridians are great circles through both poles, and parallels are smaller circles running beside the equator. The pole and equator vocabulary is borrowed from Earth, whose real shape is not a perfect sphere, so outside astronomy it serves only as illustration.
Written as a quadratic polynomial, the sphere is a quadric, a kind of algebraic surface. Some coefficient choices give an imaginary sphere with no real points or a point sphere shrunk to one location, and when the leading coefficient is zero the equation describes a plane, which can be viewed as a sphere of infinite radius centred at a point at infinity. Spherical coordinates parameterise it with a fixed radius and angles.
Archimedes, in On the Sphere and Cylinder around 225 BC, proved that a sphere holds double the space left over between it and a snugly fitting cylinder, and derived its surface area from the fact that projecting onto that cylinder’s side preserves area. A sphere inscribed in a cube fills about 52.4 percent of it, roughly 0.524 cubic metres for a one-metre cube. Soap bubbles settle into this shape, and pressure vessels and most curved mirrors and lenses are built on it.
Source: Sphere
