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Write the area A of square as a function of its perimeter P.

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 $4cm$.

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