The math behind plier sizes within a model range

[YesIHaveAHammer]

Have you always wondered why pliers come in the sizes they do? Looks like someone chose some nice looking inch numbers, and just spaced them a bit further apart as the sizes got bigger so we don't have too many sizes. That is a method, but turns out there's can be some math reasoning behind it too. I was using some of that math recently and wondered if it appeared in the design of tools - it does. I've pieced this together, so do let us know if you can add or correct anything.

The Renard number series divides a range (say 4" to 36") into some sensibly spaced out steps using (not square) roots of 10, with the gaps between them getting progressively larger. So if you're designing a range of tools/boxes/whatever, you can use this to choose which sizes to include. You choose a variant of the series depending on how many steps you want and how fast it should ramp up. The most basic series "R5" goes 1, 1.6, 2.5, 4, 6.3, 10 - each size 1.6x the previous. You multiply your smallest size by these, so say you want your range of 5 pliers to start at 4", the following sizes will be 6.4", 10" etc. That's ramping up excessively fast, so you'd actually want to use a shallower series like R10 which only goes up 1.25x on each step on the way from 1 to 10, and just use the first 5 numbers from it.

In 1988 this series was used in the definition of water pump plier specifications in ISO 8976-1988. Using the R10 series, the standard lengths were produced for a range starting at 100mm, making the rest of the range go 125mm, 160mm, 200mm, 250mm, 315mm, 400mm, 500mm. Eight sizes. The standard also gives a +/- small acceptable variation for each size. The standard was updated in 2004, and again in 2021. It's principally the same, only size variations are now formulated as approximately 0.9x to 1.12x of the nominal lengths.

As a plier manufacturer, you'll have other considerations for sizing such as typical applications, customer feedback, and offering sizes in memorable round numbers both in mm and inches. Maybe you think a 200mm model is not so necessary (as seems to be the case for Knipex with their Cobra/Alligator models), and you'd rather offer an extra inbetween size further down in the range. You can calculate an R series for any ramp up and number of steps you want, and in this case it looks like they used R8 which goes up 1.33x on each step - giving the sizes 100mm, 133mm, 178mm, 237mm, 316mm, 422mm, 562mm, 750mm etc. - thus losing the 200mm size and re-spacing its neighbours to cover the gap. We don't need to go more than 562mm, so for that inbetween size let's also look at R16 which goes up 1.15x on each step - at the bottom end this could offer us the extra sizes 115mm, (133mm appears here too), and 154mm. We can drop the 133mm and take both 115mm and 154mm instead, getting us back to eight sizes.

With a bit of rounding to nice numbers, we end up with a range of 100mm (~4"), 125mm (~5"), 150mm (~6"), 180mm (~7"), 250mm (~10"), 300mm (~12"), 400mm (~16"), and 560mm (~22").

So that seems to be why those sizes exist, why others don't, why it doesn't mirror pipe wrenches or adjustables, etc.

ISO 8976 (R10)	R8	R8 + some R16	Nice
100	100	100	100 (~4")
125	133	115	125mm (~5")
-	-	154	150mm (~6")
160	178	178	180mm (~7")
200	237	237	250mm (~10")
250	316	316	300mm (~12")
315	422	422	400mm (~16")
400	562	562	560mm (~22")
500	750	-	-

 

[pizza]

the basis of ISO 3

 

[gizardlizard]

You have WAY too much time on your hands.

 

[Zewnten]

Thats cool I wonderd why it seemed exponential not linear

 

[JuncleJohn]

I prefer to just grab pliers that look about the size I need for the task at hand. Like most, I have several sizes and styles to choose from. Quite possibly the result of all that math you described.

John

 

[neophyte]

You have WAY too much time on your hands.

This is GarageJournal.
There is nothing wrong with OP’s post.

 

[Zewnten]

You have WAY too much time on your hands.

As does almost everyone posting on here that's kinda the point.

 

[mikey03]

Honestly this is the perfect GJ post

 

[redwrench60]

So the takeaway I’m getting here is that pliers are to be sized in increments of inches and metric simply follows our standards.

So, metric is garbage and SAE still writes the playbook! Got it!

 

[neophyte]

So the takeaway I’m getting here is that pliers are to be sized in increments of inches and metric simply follows our standards.

So, metric is garbage and SAE still writes the playbook! Got it!

I think the takeaway should be that plier sizing follows a standard mathematic curve, which explains why the standardized sizes are not usually whole inch measurements.
Usually though, plier manufacturers round the plier size to the nearest convenient whole number, or fraction, even if the pliers don’t actually match that measurement exactly.
Ie. 150mm/6inch, or 300mm/12inch.

 

[YesIHaveAHammer]

So the takeaway I’m getting here is that pliers are to be sized in increments of inches

Not quite, but you highlight my omission that they are approximately those sizes in inches - I've now added "~" by the numbers to signify this. With the US being such a big market however, there is certainly an impetus to come close enough to round numbers of inches.

and metric simply follows our standards

Ha, I know what you mean... But to be precise, I believe SAE tools are just sized by convention, rather than any actual standards like ISO/DIN/JIS etc.

With a bit of rounding to nice numbers, we end up with a range of 100mm (~4"), 125mm (~5"), 150mm (~6"), 180mm (~7"), 250mm (~10"), 300mm (~12"), 400mm (~16"), and 560mm (~22").

To be exact:

100mm (~4") 3.94"
125mm (~5") 4.92"
150mm (~6") 5.91
180mm (~7") 7.09"
250mm (~10") 9.84"
300mm (~12") 11.81"
400mm (~16") 15.75"
560mm (~22") 22.05"

 

[Dave455]

And that’s why I sometimes don’t buy European pliers.

The sizes are a perfect mathematical sequence, but disregard common practice, and hand size!

Too many manufacturers will offer 160mm, and 200mm, but one is too small and the other too big. For me, 7 inch pliers are about perfect - both in combination pliers and long nose

For a decent selection of options, 5, 6, 7, 8 and 10 inch cover it!

Knipex generally offer sensible options, but not always!

 

[redwrench60]

Not quite, but you highlight my omission that they are approximately those sizes in inches - I've now added "~" by the numbers to signify this. With the US being such a big market however, there is certainly an impetus to come close enough to round numbers of inches.

Ha, I know what you mean... But to be precise, I believe SAE tools are just sized by convention, rather than any actual standards like ISO/DIN/JIS etc.

I say all this with a playful sense of humor, but what about when “convention” IS the standard? After all, I ain’t never seen a 12.7mm drive ratchet before. Does it irritate people in metric countries to have to call it a 1/2” drive ratchet?

I’m walking into the shop first thing Monday morning and asking for a pair of 300 mm Channellocks. I wonder what I’ll get besides dumb looks.

 

[YesIHaveAHammer]

but what about when “convention” IS the standard?

I guess standards can be standards either by a standards authority, or by general consensus.

After all, I ain’t never seen a 12.7mm drive ratchet before. Does it irritate people in metric countries to have to call it a 1/2” drive ratchet?

Not at all. As it's just a handful of sizes, that roll off the tongue easily, and don't need measuring or calculating, I don't think it could be improved.

Drive sizes, tools, fasteners, and the places they're used have all evolved together to suit each other, so I can't ever see that there'd be any need or benefit to reinvent drive sizes.

 

[Debcrow]

So we are saying the Meth behind tool nomenclature worldwide?

 

[1cargarage]

Wonderful to see a post like this in its natural habitat. GJ Tool Discussion is precisely the place for this kind of pedantic discussion.

For the record, I love it.

A while back, I and some other members were getting flamed by somebody on here about overanalyzing some innocuous tool feature/detail. Basically, I said to the person that THIS (GJ TOOL DISCUSSION FORUM) is precisely the place for overanalyzing tool-related arbitrarity. The thread that always comes to mind (for me) is "How do you leave a Crescent wrench?", which is 6 pages (205 responses)

Every person who clicks on this thread or the Cresecent wrench thread is guilty of what is commonly deemed "overthinking"

 

[Hannahranga]

After all, I ain’t never seen a 12.7mm drive ratchet before.

Some of the Japanese tool brands use the metric dimensions instead of 1/4,3/8,1/2 etc.
There's also plenty of "metric" sizing standards that are basically imperial with the " marks filled off see NB/DN pipe sizing

 

[Aaron_W]

And that’s why I sometimes don’t buy European pliers.

The sizes are a perfect mathematical sequence, but disregard common practice, and hand size!

Too many manufacturers will offer 160mm, and 200mm, but one is too small and the other too big. For me, 7 inch pliers are about perfect - both in combination pliers and long nose

For a decent selection of options, 5, 6, 7, 8 and 10 inch cover it!

Knipex generally offer sensible options, but not always!

It actually makes sense that "English" measurement might be more ergonomic. The math isn't as simple as metric but many measurements were based around practical measurements, how far one could walk in an hour, the length of an average stride, the size of a thumb, palm, foot etc.

 

[Hannahranga]

It actually makes sense that "English" measurement might be more ergonomic. The math isn't as simple as metric but many measurements were based around practical measurements, how far one could walk in an hour, the length of an average stride, the size of a thumb, palm, foot etc.

Now I'm curious how imperial ended up with fractional inches instead of decimal ones, I assume the whole 12 inches to a foot started it but still decimal inches doesn't seem unreasonable (and becomes the standard for machining)

 

[Aaron_W]

Now I'm curious how imperial ended up with fractional inches instead of decimal ones, I assume the whole 12 inches to a foot started it but still decimal inches doesn't seem unreasonable (and becomes the standard for machining)

They are based around halving. Many of these standards go back to the Roman empire when the average person had minimal math skills.

Just about anybody can figure out 1/2. 1/2 of 1 = 1/2, 1/2 of 1/2 = 1/4, 1/2 of 1/4 = 1/8, 1/2 of 1/8 = 1/16 etc.


A foot was literally a man's foot, an inch was the width of a man's thumb at the knuckle. There was an intermediate size palm which was the width of a man's palm (4x the width of a thumb or 4"). Smaller than an inch they used barleycorns (about 3 to an inch).

When more repeatable standards became the norm 12 is actually a useful number, easily divided by 2,3,4 and 6. 10 can only be cleanly divided by 2 and 5. Decimal systems came along much later than fractions. Zero as we know it didn't really exist until around 500AD.