Question

The circumference of a circle is $123.2cm$. Taking $\pi = \dfrac{{22}}{7}$, calculate the area of the circle in $c{m^2}$, correct to the nearest $c{m^2}$ is:

A) $196c{m^2}$
B) $1207c{m^2}$
C) $2414c{m^2}$
D) $2207c{m^2}$

Answer 1

Hint: We’ve been given the value of the circumference of the circle, using the formula of the circumference of the circle, we’ll get an equation in radius.

Solving that equation we’ll get the value of the radius of the circle, and we can easily calculate the area of the circle by substituting the value of the radius in the formula of area of the circle.

Complete step by step solution: Given data: circumference of the circle$ = 123.2cm$

Let the radius of the circle be ‘r’

We know that the circumference $ = 2\pi (radius)$

Therefore, the circumference of the given circle $ = 2\pi r$

Substituting the value of circumference from the given data

\[ \Rightarrow 123.2 = 2\pi r\]

Dividing both sides by 2

$ \Rightarrow 61.6 = \pi r$

Dividing both sides by π

\[ \Rightarrow r = \dfrac{{61.6}}{\pi }\]

Now, we know that the area of the circle$ = \pi {r^2}$

Now, substituting the value of ‘r’

$ = \pi {\left( {\dfrac{{61.6}}{\pi }} \right)^2}$

Squaring the term

$ = \dfrac{{3794.56}}{\pi }$

substituting the value of $\pi = \dfrac{{22}}{7}$

$ = \dfrac{{\left( {3794.56} \right)7}}{{22}}$

Simplifying the numerator

$ = \dfrac{{26561.92}}{{22}}$

$ = 1207.36c{m^2}$

Option(B) is correct

Note: Alternative method for finding the radius can be

Let the diameter of the circle is d.

We know that the circumference$ = \pi (diameter)$

Therefore, the circumference of the given circle$ = \pi d$

Substituting the value of circumference from the given data

$ \Rightarrow 123.2 = \pi d$

Dividing both sides by π

\[ \Rightarrow \dfrac{{123.2}}{\pi } = d\]

Now we know that $radius = \dfrac{{diameter}}{2}$

Substituting the value of the diameter

$ \Rightarrow radius = \dfrac{{123.2}}{{2\pi }}$

$\therefore radius = \dfrac{{61.6}}{\pi }$, which is equivalent to the value of radius in the above solution.