# The Wastage of Pencil Sharpening

## ...and an unexpected use of the formula for cone volume

29 May 2020

How much of the lead in a regular HB pencil do you actually use?

My hunch was that I waste about a quarter of the lead when I'm sharpening.

But it turns out I wasn't even close.

Imagine a sharpened pencil like the one in the diagram below. Keep using the pencil until it's become blunt. Maybe that means using two thirds of the length of the nib. Then sharpen the pencil to a point, as indicated by the blue triangle.

To work out the proportion of lead that is used, calculate the volume of the 'cone' of pencil lead that is used, and divide it by the total lead in the 'cylinder' that gets sharpened.

The formula of wastage is quite elegant:

The base of the cone has a radius of 'x'. If you use a pencil until it is blunt, then x = R, the radius of the lead. But if you sharpen the pencil as soon as it has lost its point, then x is close to zero.

What are the consequences?

Even if you use up the entire nib of the pencil before sharpening, you only use 33% of the lead, wasting the other 67%. Use half the nib, and you waste 90%. Less than half, and you're wasting almost everything.

Is there a way to reduce wastage? Here are my tips:

- Don't sharpen your pencil to a sharp point, a slightly flat tip saves you a lot of lead;

- Only sharpen when you have to, because the blunter you make the pencil, the less you waste;

- If you really want to save lead, use a propelling pencil. In theory this means you waste no lead at all.

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