A marble and the Earth. Two napkin rings of the same height.
Their volumes are exactly the same.
Drill a cylindrical hole through the center of any sphere.
The remaining "napkin ring" has a volume that depends only on its height
— not on the size of the original sphere!
A ring from a ping-pong ball and one from Jupiter,
if the same height, have identical volumes.
No matter how different the sphere sizes, the napkin ring volume is:
The height h is all that matters. Sphere radius cancels out completely!
Cross-sections at every height have equal area, so volumes are equal
As the sphere gets larger, two competing effects occur:
1. The ring gets thinner — the cylinder hole must be wider to maintain the same height h, leaving less material.
2. The ring gets longer — the circumference increases with the larger radius.
These two effects exactly cancel, leaving volume dependent only on height!
If two solids have cross-sections of equal area at every height, they have equal volumes.
At height y from center, the napkin ring's cross-section is an annulus (ring shape) with area:
Substituting, all R terms cancel! The area depends only on y and h, not on R. Since cross-sections match for any sphere size, volumes must be equal.
This means a napkin ring from a basketball and one from the Sun, cut to the same height, contain the same amount of material!