Question

Find the area of circle whose circumference is $\pi $. \[\]

Answer 1

Hint: We recall the definitions of circle, radius, circumference and area. We use the formula of circumference $C=2\pi r$ to get the length of the radius $r$. We use the obtained $r$ to find the required area $A=\pi {{r}^{2}}$.\[\]

Complete step by step answer:
We know that the circle is shaped on a plane which formed as a locus of points at equal distance from the fixed point. The fixed point is called centre and the line segment joining the centre to any point of the circle is called radius. We need three points to name the circle. We show the figure of circle ABC with centre O and radius A below. \[\]

The length of outline of any shape or curve in a plane is called perimeter of the shape. The perimeter of a circle is called circumference. The circumference $C$ of circle with radius $r$ is given by the formula
\[C=2\pi r\]
 Here $\pi $ is an irrational number and a universal constant whose value is approximately $3.141$.
The area of any shape or closed curve is the amount region enclosed by that closed curve. Circle is a closed curve because the starting point and the end point for the circle are the same. The area $A$ of the circle is given by
\[A=\pi {{r}^{2}}\]
We are given the question that the area of the circle is whose circumference is $\pi $. Let its radius be $r$. So we have,
\[\begin{align}
  & C=2\pi r \\
 & \Rightarrow \pi =2\pi r \\
\end{align}\]
We divide both side of above equation by $\pi $ to have,
\[\begin{align}
  & \Rightarrow 1=2r \\
 & \Rightarrow r=\dfrac{1}{2} \\
\end{align}\]
We find the area of the circle using $r=\dfrac{1}{2}$ in the formula for area
\[A=\pi {{r}^{2}}=\pi {{\left( \dfrac{1}{2} \right)}^{2}}=\pi \times \dfrac{1}{4}=\dfrac{\pi }{4}\]

So the area of the circle is $\dfrac{\pi }{4}=0.785$ square unit. \[\]

Note: We need not to confuse between circumference of circle and semicircle whose circumference is $\pi r$. We can also use other approximations of $\pi $ like $\dfrac{22}{7}$.The line segment joining two points is called a chord and the chord passing through the centre is called diameter. Diameter is twice the radius.