Here is a "physics" "proof": Far from any gravitational forces, the napkin ring sits captured between two parallel rigid glass plates at z=r and z=-r. The height 2r cylindrical center is also made of rigid glass. However the curved outer surface is a flexible membrane and the interior of the napkin ring is filled with the appropriate volume of incompressible water. Even though the membrane is free to move (other than being fixed at the plates) the water will hold the napkin ring shape (the portion of a sphere between the plates). Why? Imagine extending the membrane to be a full sphere of radius R, filling the previously empty cylinder (and caps) with water and then having the glass disappear. The fluid will hold its spherical shape due to fluid pressure. The portion of water which was in the napkin ring won't know that anything changed, the glass exerted the same forces as the new water does along the same contour. This shows that the shape of the napkin ring is completely determined by the volume of water it encloses and not by R (so long as R>r.). Since this volume is independent of R, we might as well use R=r. QED?

Variation: Start with a volume of $4/3 \pi r^3$ of fluid in a membrane. It will naturally take a spherical shape. Imagine a long cylinder of glass (inner radius 0) passing along the axis and glass plates tangent at the poles. Now let the radius of the centeral cylinder grow. The water always thinks it is part of some bigger volume of water and keeps a spherical shape (since the glass presses back on the water with the force that the water presses on it).

EDIT: More briefly than before . Here is a naive physical argument which might meet the request of a point of view from which the result would immediately appear to be just what one would expect before going through the argument.

• A blob of (incompressible) fluid volume V will form a spherical ball of radius (what it needs to be) if uncontrained
• A blob of fluid volume V constrained between two parallel plates at z=r and z=-r will form the height 2r central slice of a sphere of radius R where R is just right so that the slice has volume V (provided r is not too large relative to V in which case we get a sphere)
• A blob of fluid of volume V constrained between two parallel plates at z=r and z=-r and with a cylinder of height 2r and radius q imposed in the middle will form (along with the cylinder) the height 2r central slice of a sphere of radius R where R is just right so that the slice has volume $V+\pi q^2 h$. This shape might not have the curved boundry reach the cylinder
• Imagine that the volume is just right to get that napkin ring. Now start shrinking q. We will still have a napkin ring. Keep going until q=0 and we see that the volume was that of a sphere of radius r.

(read the previous version if you wish, it might not be worth it)

Here is a "physics" "proof": Far from any gravitational forces, the napkin ring sits captured between two parallel rigid glass plates at z=r and z=-r. The height 2r cylindrical center is also made of rigid glass. However the curved outer surface is a flexible membrane and the interior of the napkin ring is filled with the appropriate volume of incompressible water. Even though the membrane is free to move (other than being fixed at the plates) the water will hold the napkin ring shape (the portion of a sphere between the plates). Why? Imagine extending the membrane to be a full sphere of radius R, filling the previously empty cylinder (and caps) with water and then having the glass disappear. The fluid will hold its spherical shape due to fluid pressure. The portion of water which was in the napkin ring won't know that anything changed, the glass exerted the same forces as the new water does along the same contour. This shows that the shape of the napkin ring is completely determined by the volume of water it encloses and not by R (so long as R>r.). Since this volume is independent of R, we might as well use R=r. QED?

Variation: Start with a volume of $4/3 \pi r^3$ of fluid in a membrane. It will naturally take a spherical shape. Imagine a long cylinder of glass (inner radius 0) passing along the axis and glagg plates tangent at the poles. Now let the radius of the centeral cylinder grow. The water always thinks it is part of some bigger volume of water and keeps a spherical shape (since the glass presses back on the water with the force that the water presses on it).

Here is a "physics" "proof": Far from any gravitational forces, the napkin ring sits captured between two parallel rigid glass plates at z=r and z=-r. The height 2r cylindrical center is also made of rigid glass. However the curved outer surface is a flexible membrane and the interior of the napkin ring is filled with the appropriate volume of incompressible water. Even though the membrane is free to move (other than being fixed at the plates) the water will hold the napkin ring shape (the portion of a sphere between the plates). Why? Imagine extending the membrane to be a full sphere of radius R, filling the previously empty cylinder (and caps) with water and then having the glass disappear. The fluid will hold its spherical shape due to fluid pressure. The portion of water which was in the napkin ring won't know that anything changed, the glass exerted the same forces as the new water does along the same contour. This shows that the shape of the napkin ring is completely determined by the volume of water it encloses and not by R (so long as R>r.). Since this volume is independent of R, we might as well use R=r. QED?

Variation: Start with a volume of $4/3 \pi r^3$ of fluid in a membrane. It will naturally take a spherical shape. Imagine a long cylinder of glass (inner radius 0) passing along the axis and glass plates tangent at the poles. Now let the radius of the centeral cylinder grow. The water always thinks it is part of some bigger volume of water and keeps a spherical shape (since the glass presses back on the water with the force that the water presses on it).