The question **aims** to represent the area of a square in terms of its perimeter P.

The **area of a square** is defined as the measure of the space it covered. The area of the square is found by its sides, because all the sides of a square are equal to the area of the square. **Square meters, square feet, square inches, and square inches** are typical **units** for measuring square area.

The **perimeter of the square** is basically the total length around its boundary. The perimeter of the square is represented by P. The term perimeter of a square is calculated by the summation of all of its sides. **Inches, yards, millimeters, centimeters, and meters** are typical **units** for measuring perimeter.

## Expert Answer

The **length of the side** of the square is given as $a$.

All the sides of the square are **equal.** The formula of the area of the square is given by the **square of its sides**:

\[A=a^2\]

The **perimeter** $P$ is given by the **sum of all the sides of the square**:

\[P=a+a+a+a=4a\]

**Step 1: **

**Solve** $a$ for the **formula of the perimeter.** Take the value of the side from the perimeter formula and plug it into the formula of the area of the square.

\[P=4a\]

\[a=\dfrac{P}{4}\]

**Step 2:**

**Substitute** $a$ from step 1 from the formula of the perimeter to the formula of the area.

\[A=a^2\]

\[a=\dfrac{P}{4}\]

\[A=(\dfrac{P}{4})^2\]

\[A=\dfrac{P^2}{4^2}\]

\[A=\dfrac{P^2}{16}\]

The formula of the **area of the square** in **form of its perimeter** is represented by:

$A=\dfrac{P^2}{16}$

## Numerical Result

The** formula of the area of the square** in form of its **perimeter** is represented by:

\[A=\dfrac{P^2}{16}\]

## Example

**Find** the **area of the square** if the **perimeter** is $4cm$.

**Solution:**

The** formula for the area of the square** is shown as:

\[A=a^2\]

where $a$ represents the **side of the square**.

The formula for the **perimeter of the square** is shown as:

\[P=4a\]

First, write the area of the square in terms of its perimeter and then plug the value of the perimeter.

**Step 1: **

**Solve** $a$ for the **formula of the perimeter.**

\[P=4a\]

\[a=\dfrac{P}{4}\]

**Step 2:**

**Substitute** $a$ from **step 1** from the formula of the perimeter to the **formula of the area.**

\[A=a^2\]

\[a=\dfrac{P}{4}\]

\[A=(\dfrac{P}{4})^2\]

\[A=\dfrac{P^2}{4^2}\]

\[A=\dfrac{P^2}{16}\]

The expression for the **area of the square** in terms of its perimeter is represented by:

$A=\dfrac{P^2}{16}$

Now **plug the value of the perimeter** into the formula:

\[A=\dfrac{4^2}{16}\]

\[A=1cm^2\]

The result of the **area of the square** is $1cm^2$ when the **perimeter of the square** is $4cm$.