The fall on an arm is a ceiling, not a floor
Every other piece of horizontal drain in a building has a minimum grade and no maximum worth worrying about. A trap arm has both, and the maximum is the one people are surprised by. The arm runs from the trap to the point where the vent picks it up, and if the arm drops far enough over that distance the trap stops being protected — the water level in the trap and the level in the arm separate far enough that the seal is exposed to whatever the drainage system is doing.
The allowance in wide circulation is one pipe diameter of total fall across the arm. On a 1-1/4 inch lavatory arm that is 1-1/4 inches of fall, and at a quarter inch per foot you have used it after five feet. On a 2 inch arm it is two inches, which buys eight feet. That is why the same slope produces completely different arm limits on different fixtures, and why the small fixtures are the ones that bind.
Reading the default list
The lavatory in the starting list has 30 inches of arm on 1-1/4 inch pipe. At a quarter inch per foot that is 5/8 inch of fall against a 1-1/4 inch allowance, so it has 5/8 inch spare and could run to 60 inches before it used the lot.
The tub has 60 inches of arm on 1-1/2 inch pipe. That is 1-1/4 inches of fall against a 1-1/2 inch allowance, leaving a quarter inch. It is the tightest thing in the list, and it is tight because the arm is long rather than because anything about it is unusual. At 72 inches the same arm has used the allowance exactly, and past 72 it is over.
Steeper is not safer here
The instinct on a drain is that more slope is better. On a trap arm it works against you, because the constraint is total fall rather than rate of fall. Running the tub arm at three eighths of an inch per foot instead of a quarter takes its fall from 1-1/4 inches to 1-7/8, which is past the allowance on 1-1/2 inch pipe with the arm length unchanged.
The calculator gives the slope that exactly fills the allowance on the tightest arm, which is the number to know before somebody decides to give the whole bathroom a bit more grade for luck.
Where the arm is measured from
Codes do not agree on this and it changes the answer. Some measure the arm from the weir of the trap — the top of the outlet bend, where water would spill over. Some measure from the trap outlet. Whether a vertical drop between the trap and the horizontal run counts against the arm length is also treated differently in different places, and on a kitchen sink with a deep tailpiece that is a real difference.
The page takes whatever length you give it. Getting that length right against your own code is the part that has to happen before any of this arithmetic means anything.
Questions people ask
How long can a trap arm be?
It depends on the pipe size, the slope you run it at, and the total fall your code allows across the arm. With a one diameter allowance and a quarter inch per foot, a 1-1/4 inch arm reaches 60 inches, a 1-1/2 inch arm 72 inches and a 2 inch arm 96 inches. Change the slope and every one of those moves. Several codes also cap the arm length outright by size, which is a separate limit you have to read yourself.
Why is there a maximum fall on a trap arm at all?
Because the trap seal has to stay protected until the vent picks the arm up. If the arm drops too far over its length, the water in the trap and the drain below it can separate enough for the seal to be lost to pressure changes in the system. The rule exists to keep sewer gas on the far side of that plug of water, which is also why the remedy is never a matter of judgement on site.
Does more slope help a trap arm?
No, it is the thing that uses up the allowance. Total fall across the arm is what is limited, so a steeper arm reaches the limit in a shorter distance. Doubling the slope halves the arm length you can run. This is the opposite of the instinct people bring from sizing a building drain, where the worry is usually about too little fall.
What is the vent takeoff for?
Ordering material. Add the vertical rise and the horizontal run for each fixture, add the cutting waste, and you have the pipe to buy and the fittings to count. It is a quantity, not a design. What size the vent has to be, where it may connect, and whether the arrangement is acceptable at all come out of the adopted code and the inspection.
Can I just add slope to make a long arm reach?
That is the move the arithmetic on this page is designed to catch, because it makes the fall problem worse rather than better. A long arm at a steeper grade is further over the allowance than the same arm at a gentler one. The usual answers are moving the vent connection closer to the fixture or relocating the fixture, and which of those is available on a real job is not something a page can decide.