Question

The circumference of the circle is \[11\pi\] inches . What is the area in square inches of the circle?

Answer 1

Hint:In this question, we need to seek out the area of the circle in square inches . Given that the circle has the circumference of \[11\pi\] inches . Here \[\pi\] is nothing but a mathematical constant . First we need to find the radius of the circle , we can use the formula of circumference of a circle . The formula of the circumference is given by \[2\pi r\] , where \[r\] is the radius of the circle. As given in the question the circumference of a circle is \[11\pi\] , so equating this value to the formula of the circumference which is \[2\pi r\] . Thus we obtain the radius of the circle. From the radius of the circle, we can find the area of the circle by substituting the values of radius \[r\] and \[\pi\] constant in the formula and solve it to get the answer.

Formula used :
The circumference of the circle,
\[Circumference = 2\pi r\]
Area of the circle,
\[A = \pi r^{2}\]
Where \[A\] is the area of the circle, \[r\] is the radius of the circle and \[\pi\] is a mathematical constant.

Complete step by step answer:

Given , the circumference of the circle is \[11\pi\] .
Here first we need to find the radius of the circle.
We know that the circumference of the circle is \[2\pi r\] .
In question, given that the circumference of the circle is \[11\pi\] .
So by equating \[2\pi r\] and \[11\pi\] ,
We get,
\[2\pi r = 11\pi\]
On simplifying,
We get,
\[\Rightarrow \ r = \dfrac{11}{2}\]
On further simplifying,
We get
\[\Rightarrow \ r = 5.5\]
So the radius of the circle is \[5.5\] inches.
Now we need to seek out the area of the circle.
We will use the area of the circle formula to find the area of the circle.
\[A = \pi r^{2}\]
By substituting, the value of \[r\] and \[\pi\] ,
We get,
\[A = \left( 3.14 \right)\left( 5.5 \right)^{2}\]
On simplifying,
We get,
\[A = (3.14)(30.25)\]
On multiplying,
We get,
\[A = 94.98\]
Thus the area of the circle is approximately equal to \[95\] inches.
The area of the circle is approximately equal to \[95\] inches.

Note:Mathematically, in a circle when a line a drawn from the centre to any point on the circle then that line is known as the radius of the circle similarly when a line is drawn from a end point to the other end point of the circle passing through the centre of the circle is known as diameter of a circle. In order to solve this problem ,we need to know the basic formulae for finding the area of a circle. Also, The relation between area of circle and the circumference of the circle which can be written as
 \[A = \pi\left( \dfrac{C}{2\pi} \right)^{2}\]
So, using this relation also, we can directly find the area of the circle.