Circumference is the multiplier
The ring around a pipe has area pi over four times the difference of the squares, and that factors into something more useful: average circumference times radial gap. A 4.5 inch pipe in a 6 inch hole has an average circumference of 16.49 inches and a radial gap of 0.75, giving 12.37 square inches. A 1.32 inch water service in a 3 inch hole has a bigger radial gap, 0.84, and yet the ring is only 5.70 square inches, because the circumference doing the multiplying is 6.79 instead of 16.49.
Practically: on a large pipe, tightening the hole matters. On a small one, a generous hole costs very little area. That is the opposite of the way people usually order sleeves.
Rod, packing and sealant are three different quantities
Backer rod is measured in linear feet round the middle of each ring — the five default penetrations come to 5.65 feet, which is nothing, and is why nobody plans it and everybody runs out of the right diameter. Packing is a volume: whatever the ring holds through the thickness of the wall once the sealed depth at each face is subtracted. Sealant is a much smaller volume, only the depth at the faces. Ordering by volume when you needed feet, or the other way round, is the usual failure.
What decides the rod diameter
Not this page. Rod is selected against the gap it has to stay in and against the sealant it supports, and that is a supplier figure. The widest radial gap on your list is reported so you know which one governs the order, and nothing more is claimed.
Related pages
For cartridges from a bead size rather than a volume, the caulk coverage calculator takes the linear feet this page produces. For the static gaps that are not round, foam coverage works in board feet and cans. To find out how much of a building total these rings represent, the gap survey page ranks openings by area. Where the pipe goes through a roof rather than a wall, the roof penetration flashing calculator counts boots and curbs, and window and door flashing covers a rough opening.
Questions people ask
How do I measure the outside diameter of a pipe I cannot get calipers on?
Run a tape or a strip of paper around it, read the circumference, and divide by 3.1416. It is accurate enough for this and it works on a pipe against a wall where calipers do not. Note that nominal pipe sizes are not outside diameters: half inch copper is 0.625 outside, and two inch steel conduit is 2.375. Use the measurement, not the label.
Why is the sealed depth separate from the wall thickness?
Because they are different jobs. Sealant is a surface material with a working depth; filling a six inch hole solid with it is expensive, slow and generally worse than a packed ring with a sealed face. The page therefore reports a packing volume for the middle of the wall and a sealant volume for the faces, and lets you set how deep each face is worked and whether both faces are done.
Can I use this for a hole with no pipe in it?
No, and it will skip the line. With no pipe there is no ring, just a circle, and the gap survey calculator on this site handles a plain round hole. This page exists for the case where something passes through the opening and the remaining space is an annulus.
Does the collar plate count include the fixing holes?
The plate area has the pipe hole deducted and nothing else, so it is the material in the ring of plate. Fixing holes are small enough not to matter for an area figure and the fastener count is reported separately, per plate per face, at whatever number you enter.
Is a sealed penetration proof against anything?
The page makes no such claim and neither should anyone else on your behalf. It reports geometry: how much ring there is, how much rod goes round it, and how much material fills it. What passes through what, and whether an opening is adequately closed for any purpose, are not questions a materials take-off can answer, and where animals are involved they belong with your state wildlife agency and a licensed operator.