# How do vernier calipers work?

How do the vernier calipers work? I’ve always wondered about this, since we were only taught how to use the vernier calipers in school, but never why it worked. Turns out that its takes just five minutes to understand.

It’s a beautiful idea. The calipers consists of a ruler like frame, a pair of teeth and a slide that moves on the frame (see Fig 1). One tooth is fixed at one end to the frame–the left end in Fig 1– and the other movable tooth is attached to the slide.

The frame has one set of markings (the main scale), usually graduated with divisions of size 1mm. There is another set of markings on the movable slide (the vernier scale) that lines up with the fixed scale. The vernier scale has finer divisions; in a standard basic vernier like in Fig 2., 10 divisions of the vernier scale correspond to 9 divisions of the main. The vernier divisions are marked 0 through 9 and then 0 again. This means that the divisions on the vernier are separated by 0.9mm. This pair of vernier calipers has an accuracy of 0.1mm. A priori, it’s not clear that we can measure 0.1mm with 0.9mm graduations.

When the 0 of the vernier is aligned with the 0 of the main, it’s clear that the $i\textsuperscript{th}$ division on the vernier lags the main by 0.1mm. When the movable tooth is advanced by 0.1mm, the 1st division of the vernier aligns with the main scale; advance it by 0.2mm and the second division aligns.

This means that we can measure distances as follows: read the main scale by looking at or immediately to the left of the zero of the vernier. Suppose it reads 11mm. See which division of the vernier aligns with the main scale; suppose its the 6th. Then the measurement is 11.6mm.

It is easily to generalize this simple idea to achieve arbitrary accuracy: Suppose the main scale has divisions of size $a$, and you want to achieve an accuracy of $a/m$, where $m$ is an integer. We have to match, then, $p$ divisions of the main scale with $q$ divisions of the vernier scale, where $p$ and $q$ are relatively prime integers. Then, $pa/q$ is the size of each vernier division, and the distance between a main scale division and vernier scale division is $a(q - p)/q$. In other words,

$accuracy = a\frac{q-p}{q}=a\frac{1}{m}$

Clearly, we must have $q-p = 1,~q=m$. The standard vernier in Fig 2. sets $p=9,~q=10,~a=1$ to achieve an accuracy of 0.1mm.

The vernier in Fig 1. has a main scale with divisions of size 1mm. The vernier scale has 50 divisions. They correspond to 49 divisions on the main scale. That is, we have $p=49,~q=50,~a=1$, which gives an accuracy of $1/50=0.02$mm.