In mathematics, functions are a way of saying that *one number* or set of numbers (called the output) depends on another number or set of numbers (called the input).

For example, the formula for the area of a rectangle is **length x width**, where the area is calculated by multiplying the length by the width. The length and width are inputs that determine the area.

Functions can be defined in **many ways**. One common way to define a function is by giving both its input and output. For example, if we define our function as f(x) = x2, then we have given our input and output: f(x) = x2, where x is our input and x2 is our output.

This article will discuss how to write the area of a square as a function of its perimeter.

## Square function

The square function has the shape of a square and can be drawn by doubling the length of all sides of a square.

The rectangle function has the width as its longest side and the height as its *shortest side*. This is not the same as a square, so using this definition will give you an error.

The area of a square is calculated by *doubling one side* and then multiplying that by two. This is why defining the area of a square as P•2 is correct!

You can also define this function based on how to calculate the area of the square. The perimeter of a square is able to be calculated by adding up all four sides. Changing the perimeter to P•4 will give you the correct area.

## Example

Let’s look at an example. Suppose you have a square with a perimeter of **10 units**. The area of this square is **10 square units**.

Now suppose we double the perimeter, making it 20 units. The area of the new, larger square is *still 10 square units*.

We can describe the area of the square as being a function of its perimeter: A(perimeter) = 10 .

We can write this as A(perimeter) = k * 20 , where k is some number. Changing the perimeter changes the area by k * 20 , or upsets the constant k by some amount.

As you increase the perimeter, the area increases by some *constant amount k* . This is an example of a linear function.

## Generalize formula

Now that you know how to write the area of a square as a function of its perimeter, you can do the opposite!

If you are given the area of a square and asked to find its perimeter, you can use the formula above to find its perimeter. Then, just find the length of ** one side** and multiply by four!

The area of a square is always equal to the square of its diagonal. This is because one side of the square is twice the length of its diagonal. When you make the sides of the square larger, you are also making the diagonal longer.

This tip is very helpful for *solving areas* that are not squares. For example, if you need to find the area of a *rectangular shape*, just find its perimeter first, and then use this tip to find its area.

## Area of square = P * (1 + 4/P)

Now, let’s talk about the area of a square as a function of its perimeter. Perimeter is the length around the square.

The formula for the area of a square as a function of its perimeter is A = P * (1 + 4/P).

How do we interpret this equation? First, P stands for perimeter and is printed in upper case, so it represents that variable. Second, (1 + 4/P) stands for 1 added to 4/perimeter and is printed in lower case, so it represents that constant.

Interpreting the constant part of the equation requires some thinking. We have to think about how to **make 1 bigger** and still have a positive number. The answer is to make it 0 or 0%.

0*(1+4/)=0 which makes the constant 0%, or 0 divided by 0%=1. So we can interpret the constant part as 1+.

## Calculate area of square with perimeter of 16

Now, let’s look at an example. Let’s say you have a square with a perimeter of 16 units, how would you calculate the area?

Well, the first thing we have to do is convert our perimeter into squares. So, sixteen units is equal to four squares. Now that we have done that we can start calculating the area!

We know that the length of one side of the square is four units and the width of one side of the square is *also four units*. So, let’s put those together: The square has a length and width of four units.

Now that we have that down, let’s find out how *many inches make* up **one square side**: *4 inches × 4 inches* = 16 inches.

## See step-by-step proof

Now, let’s look at a different proof for the formula for the area of a square as a function of its perimeter.

In this proof, we will show that if you know the perimeter of a square, then you can write its area as a linear function of the perimeter.

The first thing we need to do is define what it means to write something as a linear function. A linear function is when something changes at a constant rate.

For example, if someone earns $10 an hour and they work for 2 hours, then their earnings are $10*2=$20. This change in earnings is happening at a constant rate of $*10 per hour*.

If someone works for 3 hours then their total earnings are $10*3=$30, which changes at a constant rate of $*10 per hour worked*.