Question

If the area of a circle is 154\[c{{m}^{2}}\], then its circumference in cm is –

Answer 1

Hint: We need to use the relation between the area of the circle and the circumference of the same circle. We can relate them using the radius of the circle which can be solved from the given area of the circle to solve the circumference of the circle.

Complete step-by-step answer:
The circle is a curved line segment joined by points equidistant from a given point. The circumference of the circle is the length of this curved line that bounds a closed area. The area of the circle is the area bound by this curved line that forms the circle.
  

The constant distance from the center to the curved line is called the radius of the circle. We can find the circumference of a circle using the formula as given below –
\[C=2\pi r\]
Where, r is the radius of the circle,
\[\pi \] is a constant with the value of a \[\dfrac{22}{7}\].
The area for any closed surface is proportional to the square of the length involved in the shape. For a circle, the area is given by a similar formula which is related through the constant pi as –
\[A=\pi {{r}^{2}}\]
We are given the area of a circle. From the information we can find the radius of the circle as –
\[\begin{align}
  & A=\pi {{r}^{2}} \\
 & \Rightarrow r=\sqrt{\dfrac{A}{\pi }} \\
 & \text{given,} \\
 & A=154c{{m}^{2}} \\
 & \Rightarrow r=\sqrt{\dfrac{154}{\pi }} \\
 & \Rightarrow r=\sqrt{\dfrac{\dfrac{154}{22}}{7}} \\
 & \therefore r=7cm \\
\end{align}\]
The circumference of the circle can be found from the radius as we have found from the above calculations. The circumference of the circle is given as –
\[\begin{align}
  & C=2\pi r \\
 & \Rightarrow C=2\pi (7cm) \\
 & \therefore C=44cm \\
\end{align}\]
The circumference of the circle is 44cm.


Note: The constant \[\pi \] was found by finding the ratio of the circumference of the circle to the radius of the circle for different circles. The mathematician was baffled by the result as he kept on getting a constant value by dividing the two values for any random circles.