Question

The circumference of a circle is equal to the side of a square whose area measures 407044 sq. cm. What is the area of the circle?
Is it possible to solve this question using the digital root method?
A) 22583.2 sq. cm
B) 32378.5 sq. cm
C) 41263.5 sq. cm
D) 394834 sq. cm
E) Cannot be determined

Answer 1

Hint:
From the given area of the square, find the side of the square, using the formula $A = {\left( {{\text{side}}} \right)^2}$. Next, use the side of the square as the circumference of the circle and calculate the radius of the circle. For, the calculated radius of the circle, find the area of the circle using the formula, $A = \pi {r^2}$

Complete step by step solution:
As the area of the square is ${\left( {{\text{side}}} \right)^2}$ and we are given that the area of the square is 407044 sq. cm.
Thus, calculate the side of a square.
$
  {\left( {{\text{side}}} \right)^2} = 407044 \\
  {\text{side}} = \sqrt {407044} \\
  {\text{side}} = 638{\text{cm}} \\
 $
Therefore, the circumference of the circle will be 638 cm.
That is, $2\pi r = 638$
Determine the radius of the circle by solving the equation.
$
  2\left( {\dfrac{{22}}{7}} \right)r = 638 \\
  r = \dfrac{{638\left( 7 \right)}}{{2\left( {22} \right)}} \\
  r = 101.5{\text{cm}} \\
 $
Calculate the area of the circle using the formula, $A = \pi {r^2}$
$
  A = \left( {\dfrac{{22}}{7}} \right){\left( {101.5} \right)^2} \\
  A = \left( {\dfrac{{22}}{7}} \right)\left( {101.5} \right)\left( {101.5} \right) \\
  A = 32,378.5{\text{ sq}}{\text{.cm}} \\
$
Hence, option B is correct.
In the digital root method, we add the digits of the number and continue the process till a single digit is left. The single digit that we get is the digital root of the number.
It is used for faster calculations. For example, if we want to check whether a number is divisible by 3 , we find its digital root. If the digital root is divisible, then the whole number will be divisible by 3.
The digital root method is not applicable for this question.

Note:
The area of the square is $A = {\left( {{\text{side}}} \right)^2}$. The circumference of the circle is the boundary of the circle given by $2\pi r$, where $r$ is the radius of the circle. The area of the circle is given by $A = \pi {r^2}$. The area is always calculated in square units.