The purpose of this question is to explain the geometry of the circle, understand how to calculate the circumference and the area of the circle, and learn how the different formulas of the circle relate to each other.
The assemblage of points that are at a specified distance $r$ from the center of the circle is called the circle. A circle is a closed geometric shape. Examples of circles in everyday life are wheels, circular grounds, and pizzas.
The radius is the distance from the center of the circle to a point on the boundary of the circle. The radius of the circle is denoted by the letter $r$. The radius $r$ plays a vital role in the formation of the formulas of the area and circumference of the circle.
A line whose endpoints lie on a circle and pass through the center is called the diameter of a circle. The diameter is represented by the letter $d$. The diameter is twice the radius of the circle, that is $d = 2 \times r$. If the diameter $d$ is given, the radius $r$ can be calculated as $r = \dfrac{d}{2}$.
The space occupied by the circle in a two-dimensional plane is called the area of a circle. Alternatively, the area of the circle is the space occupied within the boundary/circumference of the circle. The area of the circle is denoted by the formula:
\[ A = \pi r^2\]
Where the $r$ denotes the radius of the circle. The area of the circle is always in the square unit, for example, $m^2, \space cm^2, \space in^2$. $\pi$ is a special mathematical constant and its value is equal to $\dfrac{22}{7}$ or $3.14$. $\pi$ denotes the ratio of the circumference to the diameter of any circle.
Circumference is the length of boundary of the circle. The circumference is equal to the perimeter of the circle. The length of the rope that tapes around the circle’s border absolutely will be equal to its circumference. Formula to calculate the circumference is:
\[ C = 2 \pi r\]
Where $r$ is the radius of the circle and $\pi$ is a constant equal to $3.14$.
Expert Answer
The area of a circle is:
\[ A = \pi r^2 \]
The circumference of a circle is:
\[ C = 2 \pi r \]
Now making radius $r$ the subject in the circumference equation:
\[ C = 2 \pi r\]
\[ r = \dfrac{C} {2 \pi} \]
Inserting the $r$ in the equation of Area $A$:
\[ A = \pi r^2 \]
\[ A = \pi (\dfrac{C} {2 \pi})^2 \]
\[ A = \pi (\dfrac{C^2}{4 \pi^2}) \]
\[ A = \cancel{ \pi} (\dfrac{C^2}{4 \cancel{ \pi^2}}) \]
\[ A = \dfrac{C^2}{4 \pi} \]
Numerical Answer
Area $A$ of a circle as a function of its circumference $C$ is $\dfrac{C^2}{4 \pi}$.
Example:
Calculate the area if the radius of the circle is $4$ units.
\[ A = \pi r^2 \]
\[ A = 3.14 (4)^2 \]
\[ A = 50.27 \]