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.

\[ A = \pi r^2 \]

\[ A = 3.14 (4)^2 \]

\[ A = 50.27 \]