The length and breadth of a rectangular sheet are$16.2cm$ and $10.1cm$ respectively. The area of the rectangular sheet in appropriate significant figures and error is
$A.164\pm 3c{{m}^{2}}$
$B.163.62\pm 2.6c{{m}^{2}}$
$C.163.6\pm 2.6c{{m}^{2}}$
$C.163.62\pm 3{{m}^{2}}$

Hint: here the significant figures and error is to be calculated. In error absolute error is to be calculated first and then convert it into percentage error. Rounding off of the values is also done in accordance with the law.

Complete step by step solution:
First of all let us calculate the error, in the product of two quantities $a$and\[b\],
$x=a\times b$\[A=A\pm \Delta A=\left( 164\pm 3 \right)c{{m}^{2}}\]
Suppose $\Delta a$ is the absolute error in measurement of$a$.
\[\Delta b\] The absolute error in the measurement of\[b\]is.
Then let us take \[\Delta x\]is the error in the product of the quantities.
Then the maximum fractional error in the product is given by
\[\dfrac{\Delta x}{x}=\pm \left( \dfrac{\Delta a}{a}+\dfrac{\Delta b}{b} \right)\]
From this the percentage error in x is equal to sum of the percentage error of $a$and\[b\].
In accordance to the problem, length is
\[l=\left( 16.2\pm 0.1 \right)cm\]
Breadth is \[b=\left( 10.1\pm 0.1 \right)cm\]
Hence the area of rectangular sheet is
\[A=l\times b=\left( 16.2 \right)\times \left( 10.1 \right)=163.62c{{m}^{2}}\]
According to rule, as there is three significant figures in area and one in error, we can round off to get error=\[A=164c{{m}^{2}}\]
Let us calculate the relative error,
\[\dfrac{\Delta A}{A}=\dfrac{\Delta l}{l}+\dfrac{\Delta b}{b}\]
\[=\dfrac{0.1}{16.2}+\dfrac{01}{10.1}\]
\[=\dfrac{1.01+1.62}{16.2\times 10.1}=\dfrac{2.63}{163.62}\]
Therefore relative error in area will be
\[\Delta A=A\times \dfrac{2.63}{163.62}c{{m}^{2}}=2.63c{{m}^{2}}\]
In order to make it only one significant figure round off is done,
\[\Delta A=3c{{m}^{2}}\]
Therefore area can be written as,
\[A=A\pm \Delta A=\left( 164\pm 3 \right)c{{m}^{2}}\]

So the correct option is, option A.

Note: Significant figures are also known as significant digits or precision. It is the numbers or digits which are written in a particular way or notation that is carrying meaningful contributions to its measurement resolution. If by calculations which is performed to a greater precision than that of the original measurements are reported to have a greater precision than the measurement with help of equipment supports.