What is the formula of Vernier constant?

Hint: In order to solve this question we need to understand first what Vernier caliper is. Vernier Caliper is a physical instrument used to measure length, breadth and height of any small physical cube, beaker etc. Vernier constant or the least count is the smallest measurement that a Vernier caliper could measure.

Complete step by step answer:
First let us study the structure of the Vernier caliper. The Vernier caliper has two divisions incarnated on it one is main scale division and other is Vernier scale division.Now to measure something we can use the claws or tail head of Vernier.Vernier scale division are such that upon bringing the two claws together there are some Vernier divisions coincide with main scale division.

So a Vernier constant is defined as $V.C = 1M.S.D - 1L.S.D$. Where $M.S.D$ is the main scale division and $L.S.D$ is least or Vernier scale division. Let us suppose on bringing the claws together $(n)L.S.D$ coincides with $(n - 1)M.S.D$.
So $1L.S.D = \dfrac{{n - 1}}{n}M.S.D$
Putting the values in formula of Vernier constant we get
$V.C = 1M.S.D - 1L.S.D$
$\Rightarrow V.C = (1 - \dfrac{{n - 1}}{n})M.S.D$
$\therefore V.S.D = (\dfrac{1}{n})M.S.D$

Hence, the formula of Vernier constant is $V.S.D = (\dfrac{1}{n})M.S.D$.

Note:It should be remembered that the lowest count of any device is the instrument error that we can correct by applying special measures at manufacturing. Apart from this every device has some inherent error or uncertainty which cannot be removed completely rather it is minimized. Also zero error associated with least count. This zero error is either positive or negative depending upon the relevant position of zero of Vernier division to the zero of main scale division.