# How Vernier Scale Works

## An Intuitive Understanding

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Before the days of digital calipers with LCD display, one learns how to read the Vernier scale on a metal caliper directly. There are still tutorials teaching how to read such a metal caliper correctly.

Once you learn how to use the aligned number on the Vernier scale to read out sub-millimeter, have you wondered why does it work like that?

To gain an intuitive understanding, let’s start with the zero position of a caliper (with sliding Vernier scale in yellow):

If you look at the 50th big division mark (marked as 0 on the right, after the number 9) on the yellow Vernier scale, it matches the 49mm mark on the main scale. In other words, it’s 1mm short of 50mm mark. Because this Vernier scale has 50 (5 x 10) total division marks, this 1mm shortage is evenly distributed among them, i.e. each Vernier division is 0.02mm (1/50) shorter than the 1mm it represents (50mm mapped into 50 divisions).

Thus, instead of marking 0, 5, 10, …, 50 mm on the Vernier scale, which would be just like the main scale, the marking 0, 1, 2, 3 indicates the total shortage in mm. For example, 5 mark on the Vernier scale occupies 25 divisions, since each division is 0.02mm shorter, that 5 indicates 0.5 mm shortage.

With this understanding, now why does the following read: d = 20.36mm?

1. 0 mark on the Vernier scale has surpassed 20mm, but has not reached 21mm yet, so we need to read sub-millimeter using the Vernier scale.
2. 38mm mark on the main scale aligns with 3.6 mark on the Vernier scale
3. Ignore the 3.6 number for the time being, let’s count how many total divisions on the yellow scale: it is 18
4. Here is a simple equation showing the relationships:

20mm + x + Vernier(18) = 38mm

So if each Vernier division maps to exactly 1mm, 20 + 18 = 38mm, our x would be zero. Again number 18 comes from 18 division marks on the Vernier scale.

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memento of electronics and fun exploration for my future self