Question

The radius of a disc is $1.2\;cm$. Its area according to the concept of significant figures will be given by:
A). $4.5216\;cm^2$
B). $4.521\;cm^2$
C). $4.52\;cm^2$
D). $4.5\;cm^2$

Answer 1

Hint: The area of the disc can be calculated by treating it as a circle. What is more important is to understand the significance of the digits in the number that you obtain for the area of the disc. Remember that your result should be rounded off and clipped to the same number of significant digits as in the radius.

Formula Used:
Area of a disc $A_{disc} = \pi r^2$, where r is the radius of the disc.

Complete step-by-step solution:
Let us begin by understanding the concept of significant figures.
As the name suggests, significant figures are the digits of a number that carry some importance towards the value of the number. This “importance” can physically be described as the degree of accuracy that the digit lends to the number. It determines if a digit imparts the value to the number taking part in a measurement of a calculation.
There are a few rules for identifying significant figures:
All non-zero digits are considered significant.
(For example, 52 has 2 significant figures, 25.57 has 4 significant digits)
Zeros appearing in between two significant figures are significant
(For example, 504.66 has 5 significant figures)
Zeros to the left of the significant figures (leading zeroes) are not significant
(For example, 0.0034 has 2 significant figures)
Zeroes to the right of the significant figures (trailing zeroes) are significant only if they are to the right of the decimal point.
(For example, 1.20 would have 3 significant figures, whereas 120.00 would have 5 significant zeros)
Now, let us look at the problem dealt with us.
We are given the radius of the disc as $r = 1.2\;cm$.
Note that this is expressed in terms of 2 significant figures.
Therefore, the area of the disc will be $A_{disc} = \pi r^2 = \pi \times 1.2^2 = \pi \times 1.44 = 4.524\;cm^2$
But, we need the area in the same number of significant figures as the radius posed to us.
Therefore, with the demanded consistency in the number of significant digits, the area of the disc will be $4.5\;cm^2$.
Therefore, the correct option is C. $4.5\;cm^2$

Note: It is important to remember that though finding the result of a measurement or a calculation might initially seem more precise when calculated to a lot of decimal places, this is not necessarily the case. The introduction of spurious digits or the digits introduced by calculations carried out by a greater precision than that of the original data, or the measurements inferred to a greater precision than that supported by the instrument are considered to be non-significant as this does not help in contributing to the accuracy of the system in any way as the maximum threshold for this accuracy is curbed by the number of significant digits in the original data or measurements.