I don’t know why 0.5mm pencils are more popular than 0.7mm and 0.9mm. I had 0.5mm pencils around the house, but when I saw my son break the lead three times in 15 seconds I decided to try 0.7mm. Much less breakage. At the time I didn’t even know that you could get 0.9mm pencils, but when I found out, I got one. I was afraid that they would be too thick, but in fact the lines they produce doesn’t really seem much thicker — it’s just smoother to write them. That’s probably because the worn surface of the lead is curved, and for the same amount of pressure a 0.9mm lead will penetrate less deeply into the paper. That reduces both the spot size and the drag of the pencil.

The big win, though, is that 0.9mm leads are much less prone to breakage than 0.5mm. I’ll spare you the math, but at a first approximation the strength of the lead goes up with the cube of the diameter, so according to the simple model I used (I make no claims to either accuracy or precision) it takes something like 6 times as much force to break a 0.9mm lead as a 0.5mm. Also, because the 0.9mm lead is not making as sharp a trough in the paper, there is less perpendicular force on the lead. I’ve been using 0.9mm pencils for a year or so now, and I have yet to break a lead. This diagram shows their relative sizes:

Other advantages to using thicker lead:

– You don’t have to advance the lead as often.

– You can make a bolder line, if you want to. I do a lot of underlining and quick diagrams, and bolder lines work well for both.

– The mechanism seems less finicky. I used to have to disassemble 0.5mm pencils from time to time, but I never do that anymore.

– Lines are more easily erasable because the lead hasn’t made so sharp a trough in the paper.

Now I can understand that for somebody doing drafting, being able to make a really fine line is important. Such folk may even need a 0.3mm pencil. But if you are just doing homework or taking notes, you might want to try a 0.9mm.

By the way, I think it would be a really cool high school science experiment to empirically determine the relative strengths of leads of different diameters.

And you might be curious why the strength goes up with the cube rather than the square of the diameter (or radius). Clearly the cross-section of the lead, which must be pulled apart, is a disk, the area of which increases with the square of the diameter. If you were just trying to pull the lead apart by pulling the two ends in opposite directions, the strength would indeed increase with the square of the diameter. But while writing, a lead breaks in response to a perpendicular force — effectively making a lever — and the cohesion of the lead has more leverage the farther away it is from the fulcrum. A larger-diameter lead adds surface area mostly on the side furthest from the fulcrum, where it does the most good.

## 6 Comments

This is what happens when engineers have insomnia. :^)

Thanks for the post! Interesting reading and helpful. I’m curious to understand if there is any proof on how much more force is required to break a .9mm vs. a .5mm. And how to estimate how much downward force the average person applies. I also find .5mm breaks way too often for me.

James, I don’t know of any proof — sorry. I came up with my estimate by assuming that it is not the downward force that causes the break but rather the perpendicular force when you drag the lead across the paper, which causes the lead to function as a lever with the breaking point as the fulcrum. Then I determined the force involved as a function of the radius so that I could see how the force increases with the radius. Of course in practice materials don’t behave like models, so what I did was *very* simplistic.

Why does 0.7mm break a lot? Should I switch to 0.9mm?

Hi Mike. If 0.7mm leads break on you a lot, 0.9mm would definitely help, but there are other things to consider as well. If you are using a cheap mechanical pencil, it may not be rigid enough, and the flexing could be causing the lead to break. You might try to cultivate a lighter touch. If the paper is not on a hard surface (e.g. if it is just the top piece of paper on a pad), then the pencil could be sinking in further, making it more prone to breakage. If the paper is not on a smooth surface, even tiny bumps put pressure on the lead. Some leads are higher quality than others. You might need to hold the pencil more upright, so that it is more perpendicular to the paper.

What are your k and x values on that equation you linked? I’m curious. Doing a little research of my own haha. I’m assuming k is the radius, and x is the length of the lead protruding from the tip of the pencil?