(Ack. The old thread referenced above where I did the calcs with the graphics was wrong - I switched the captions for the 2 diagrams, and it went downhill from there where I used the wrong numbers for the small and medium calculations. Also applied rounding incorrectly. Nobody questioned it in the 4½ years since. Hope I didn't mash this analysis up again 
)
>>I use the USPS medium triangle box to ship, max is 59" to zone 5-9, 62" for 1-4 zones, I sell hundreds
>>a rods a year so I know these for sure.
I can't arrive at either of your numbers any way I try to dice and slice it. Based on what calculations?
That Z1-4 62" number is incorrect - until 1/27/19 balloon rating applies, and 84" max - 17" girth = 67" max length before balloon rates would kick in. To arrive at 62" max length the girth would have to be 7"x7"x8" which it clearly is not for an O-1098M tube. (girth numbers are not rounded before or after adding together, but even that wouldn't account for the numbers there)
And I can't come up with anything even close to the 59" zone5-9 number: are you computing tubular package volume using the rectangular formula instead of the non-rectangular formula, or using the right formula with the wrong width and height values?
What are the outside dimensions of the USPS O-1098 tubes you have?
────────────────────────────────────────────────────────
Going to split remainder of my diatribe into a separate post to follow.
Max lengths for USPS O-1098M & O-1098S Triangular Tubes to stay under 1728in³ threshold and avoid dimensional weight issues.
The rectangular volume formula LxWxH does not apply to triangular or round tubes.
https://pe.usps.com/text/qsg300/Q120.htm#ep1023748
Non-rectangular USPS Dimensional Weight formula is L x W x H * 0.785
2018 DMM 223.1.6 (2018) 1.6 Dimensional Weight Price for Low-Density Parcels to Zones 5-9 Postage for parcels addressed for delivery to Zones 5-9 and exceeding
1 cubic foot (1,728 cubic inches) is based on the actual weight or the
dimensional weight (as calculated in 1.6.1 or 1.6.2), whichever is greater.
1.6.1 Determining Dimensional Weight for Rectangular Parcels Follow these steps to determine the dimensional weight for a rectangular parcel: a. Measure the length, width, and height in inches. Round off (see 604.7.0) each measurement to the nearest whole inch. b. Multiply the length by the width by the height. c. If the result exceeds 1,728 cubic inches, divide the result by 194 and round up (see 604.7.0) to the next whole number to determine the dimensional weight in pounds.
1.6.2 Determining Dimensional Weight for Nonrectangular Parcels Follow these steps to determine the dimensional weight for a nonrectangular parcel: a. Measure the length, width, and height in inches at their extreme dimensions. Round off (see 604.7.0) each measurement to the nearest whole inch. b. Multiply the length by the width by the height. c. Multiply the result by an adjustment factor of 0.785. d. If the final result exceeds 1,728 cubic inches, divide the result by 194 and round up (see 604.7.0) to the next whole number to determine the dimensional weight in pounds.
(that is the 2018 DMM as the 2019 isn't available yet, but I expect no changes to the above)
USPS O-1098M Medium Tube Max length:
Based on outside dimensions 6¼ x 5¼ x 5¼ which will round off to c=6" b=5" a=5" when needed for the final calc, and Girth=16.8"
Using Heron's Formula, for a triangle with sides 6¼", 5¼", and 5¼"
s=(a+b+c)/2 = (6¼+5¼+5¼) / 2 = 8.375" and
Area = √(s(s-a)(s-b)(s-c) = √[8.375(8.375-5.25)(8.375-5.25)(8.375-6.25)]
= 13.18in²
A=½(BxH) and B=b and H=h →
h = 2A/b → h = 2(13.18in²)/5.25" = 5.02"
(matches actual measurement of the O-1098M tube height I have here)
Now we use rounded off dimensions for the final USPS calc: H=h=5" and W=b=5"
L(max) = 1728 / (W * H * 0.785) = 1728 / (b * h * 0.785) = 1728 / (5" * 5" * 0.785) = 88.0"
Plugging back into the non-rectangular Dim Weight formula, 88" x 5" x 5" * 0.785 = 1727in³ which is < 1728in³
Thus an O-1098M Medium Triangular tube up to 88" long is not subject to USPS Dim Weight consideration at all.
(that actually goes up to 91" max if we choose the longest side as the "base" in the calculation, which may be USPS's intention based on interpreting the guitar example in the diagram in QSG300 referenced above)
────────────────────────────────────────────────────────
USPS O-1098S Small Tube Max length:
Based on outside dimensions 6 x 6 x 6 which will round off to c=6" b=6" a=6" when needed for the final calc, and Girth=18.2"
Using Heron's Formula, for a triangle with sides 6", 6", and 6"
s=(a+b+c)/2 = (6+6+6) / 2 = 9" and
Area = √(s(s-a)(s-b)(s-c) = √[9(9-6)(9-6)(9-6)] = 15.59in²
A=½(BxH) and B=b and H=h →
h = 2A/b → h = 2(15.59in²)/6" = 5.2"
(matches actual measurement of the O-1098M tube height I have here)
Now we use rounded off dimensions for the final USPS calc: H=h=5" and W=b=5"
L(max) = 1728 / (W * H * 0.785) = 1728 / (b * h * 0.785) = 1728 / (6" * 5" * 0.785) = 73.38" = 73"
Plugging back into the non-rectangular Dim Weight formula, 73" x 6" x 5" * 0.785 = 1719in³ which is < 1728in³
Thus an O-1098S Small Triangular tube up to 73" long is not subject to USPS Dim Weight consideration at all.
────────────────────────────────────────────────────────
The above max lengths are true today, will be true after 1/27/19 when new USPS rates go into effect*, and will also be true in 6/23/19 - they are dependent only on geometry and the 1728 in² USPS threshold.
Starting 1/27/19, balloon weight considerations will be a thing of the past, so not going to bother with those computations.
*A beneficial quirk of the changes on 1/27/19 and the deferment of the dimensional changes, is that between 1/27/19 and 6/23/19, there will be no balloon rates or dimensional rates to zones 1-4 for Priority Mail. That means the max numbers I computed above are actually not maxima for PM triangular tubes shipping to Z1-4 destinations during that period. The only constraint on those would be L+G needing to remain under the Priority Mail 108" L+G maximum:
O-1098M Girth=16.8" → 108" Max L+G - 16.8" = 91" → Zone 1-4 Max length = 91" during quirk window
O-1098S Girth=18.2" → 108" Max L+G - 18.2" = 89" → Zone 1-4 Max length = 89" during quirk window
Starting 6/23/19 the maximum PM O-1098 triangular tube lengths drop back to 88" for M and 73" for S for all zones.
