A timing belt's length is fixed by its tooth count, and closed loops only come in the counts manufacturers make. So the useful question is not only how long a belt you need, but which stock belt you can design around.
Where the formula comes from
A belt around two pulleys is two straight runs plus the arcs where it wraps. With equal pulleys, the two wrap arcs add up to exactly one full circumference.
A pulley's pitch diameter is its tooth count times the belt pitch divided by pi, so pi times the diameter is just teeth times pitch. That cancellation is why the formula is so clean: the length is twice the centre distance plus the tooth count times the pitch.
With unequal pulleys the straight runs tilt slightly and there is a small correction term, which the calculator includes. On typical printer geometry it is a fraction of a millimetre.
Open belt or closed loop
Most cartesian printers use open belt clamped at both ends to the moving carriage, because it is simpler to tension and to replace. Length is then just the path plus clamp allowance, and you cut it from a reel.
Closed loops are used where the belt has to run continuously around a loop — core XY machines, and belt-driven Z. There the tooth count has to match a product that exists.
Stock closed loops in GT2 are commonly available in steps of a few teeth, so the centre distance figure the calculator gives is what makes the design buildable.
Tension, and why a tensioner is not optional
Belt tension affects print quality directly. Loose belt shows as rounded corners and ringing; overtightened belt loads the motor bearings and can pull a frame out of square.
The right tension is firm enough that the belt does not deflect much under a finger, and no more. Some people pluck the belt and tune it to a frequency, which is repeatable if not precise.
Either way it needs an adjuster. A frame built to the exact calculated centre distance with no adjustment leaves you relying on manufacturing tolerance, which is not a plan.
Pitch is not the same as profile
GT2 and 2GT are both 2 mm pitch but the tooth profiles differ slightly, and both differ from the older T2. They are frequently interchangeable in practice and occasionally not.
MXL is 2.032 mm — an imperial pitch of 0.08 inches — which is close enough to 2 mm to look identical and far enough off to skip teeth over a long run.
Match the belt profile to the pulley profile. Mixing them wears both quickly, and the failure shows up as position loss rather than as an obvious mechanical fault.
Steel cord versus fibreglass
Cheap belts use fibreglass cords, which stretch measurably under the tensions a printer runs. Steel-corded belt stretches far less and holds calibration.
Stretch matters because it changes steps per mm slowly over time, so a machine that was dimensionally accurate drifts. It is a common and hard-to-diagnose cause of parts that used to fit.
Steel-corded GT2 costs a little more per metre and is the standard recommendation for anything that has to stay accurate.