Question

What is the area of a circle in terms of circumference?

Answer 1

Hint: We need to find the area of a circle in terms of the circumference of a circle. We start to solve the given question using the formulae of the area and circumference of the circle. Area is $\pi {{r}^{2}}$ and circumference is $2\pi r$ . We substitute the value of r in the formula of the area as $\dfrac{circumference}{2\pi }$ to get the desired result.

Complete step-by-step answer:
We are asked to find the area of a circle in terms of circumference. We will be solving the given question using the area and circumference formula of a circle.
A circle is a two-dimensional figure in which a set of points are at a fixed distance from a center. The diameter is a line that divides a circle into two halves and is twice the radius of the circle.
The circumference of a circle is the perimeter of a circle. It is usually the linear distance around the circle.
It is usually denoted by the variable C.
The circumference of the circle is given as follows,
$\Rightarrow C=2\pi r$
Here,
C is the circumference of the circle
$\pi$ is the mathematical constant
$r$ is the radius of the circle
The area of the circle is the space or the region enclosed inside the circle.
It is usually denoted by A
It is usually denoted by the variable C.
The area of the circle is given as follows,
$\Rightarrow A=\pi {{r}^{2}}$
Here,
A is the area of the circle
$\pi$ is the mathematical constant
$r$ is the radius of the circle
Now,
$\Rightarrow C=2\pi r$
Finding the above of r in the above equation, we get,
$\Rightarrow r=\dfrac{C}{2\pi }$
We need to substitute the value of r in the formula of area.
Substituting the value of r in the area of the circle, we get,
$\Rightarrow A=\pi {{r}^{2}}$
$\Rightarrow A=\pi {{\left( \dfrac{C}{2\pi } \right)}^{2}}$
Simplifying the above equation, we get,
$\Rightarrow A=\dfrac{\pi \times {{C}^{2}}}{4\times {{\pi }^{2}}}$
Cancelling out the common factors, we get,
$\therefore A=\dfrac{{{C}^{2}}}{4\pi }$

Note: If we want one equation in terms of the other, we should first search for a common variable in both equations. Then, we should solve for a common variable in the first equation and replace the value of the common variable in the second equation.