Question

The sum of the circumference and the radius of a circle is 51cm then, find the radius of the circle.

Answer 1

Hint: We solve this problem by using the formula of the circumference. Let us take the rough figure of the circle as shown below

The formula of the circumference of the circle of radius \['r'\] is given as
\[\Rightarrow C=2\pi r\]
Then we use the given condition that the sum of the circumference and the radius is equal to 51cm to find the value of the radius of the circle assuming that \[\pi =\dfrac{22}{7}\]

Complete step-by-step solution
We are given that the sum of circumference and the radius of a circle is 51cm
Let us assume that the radius of the circle as \['r'\]
We know that the formula of the circumference of the circle of radius \['r'\] is given as
\[\Rightarrow C=2\pi r\]
By using this formula to the given condition then we get
\[\begin{align}
  & \Rightarrow 2\pi r+r=51 \\
 & \Rightarrow r\left( 2\pi +1 \right)=51 \\
\end{align}\]
We know that the value of \[\pi \] that is
\[\Rightarrow \pi =\dfrac{22}{7}\]
Now, by substituting this value of \[\pi \] in above equation we get
\[\begin{align}
  & \Rightarrow r\left( 2\left( \dfrac{22}{7} \right)+1 \right)=51 \\
 & \Rightarrow r\left( \dfrac{44}{7}+1 \right)=51 \\
\end{align}\]
Now, by adding the terms using the LCM we get
\[\begin{align}
  & \Rightarrow r\left( \dfrac{51}{7} \right)=51 \\
 & \Rightarrow r=7cm \\
\end{align}\]
Therefore the radius of the given circle is 7cm.

Note: Students may get confused at the value of \[\pi \]
Here, we have the equation as
\[\Rightarrow r\left( 2\pi +1 \right)=51\]
We usually take the value of \[\pi \] as 3.14. So, let us take the value of \[\pi \] as 3.14 then we get
\[\begin{align}
  & \Rightarrow r\left( 2\left( 3.14 \right)+1 \right)=51 \\
 & \Rightarrow r=\dfrac{51}{7.28} \\
 & \Rightarrow r=7.0054 \\
\end{align}\]
Here, we can see that the radius of circle we get is 7.0054 which when rounded off to nearest number we get 7
So, we can conclude that the radius of the circle is 7cm. Here, whatever the value of \[\pi \]we take that is either 3.14 or \[\dfrac{22}{7}\] both lead to the same answer but one value gives the more precise value than the other. But we usually consider the rounded-off numbers. So, when it comes to rounding off the numbers we get the same value in both cases.