Area of a Square

The area of a square is an important concept in geometry. It allows us to understand the properties of a square even further and also opens a wide range of applications and problems involving squares. In the past, we’ve learned that squares are quadrilaterals with equal sides, so it’s time to establish the rules for finding the area of a square.

There are different ways to calculate the area of a square such as: 1) counting one-unit squares that make up the larger square, 2) using the square’s sides, and 3) utilizing the square’s diagonals. This article covers all these methods and breaks down the formulas for each method.

This discussion covers the fundamentals of a square and its area, the important dimensions and formulas we can use to find its area, and word problems involving areas of squares. By the end of this discussion, we want you to feel confident when dealing with a square’s area.

What is the Area of a Square?

The area of a square represents the amount of area covered by a given surface enclosed by a square region. As a refresher, a square is a four-sided figure that has four identical sides. Squares also have opposite sides that are parallel and corners that form right triangles. The area of a square has defined as the product of its length and width and is measured in squared units.

The images below are examples of square-shaped surfaces. The measure of the region that covers these surfaces represents their areas.

For each surface, we can define its area by counting the smaller squares found within, the square’s dimensions, or even its diagonals – depending on what’s given. This leads to a wide range of applications and problems that we can finally solve after learning about the area of a square.

Understanding the Area of a Square

Before establishing the formula for the area of square, let’s first understand how we can visualize it. Suppose that we have a 4\text{ cm} \times 4\text{ cm} square, we can illustrate it using unit squares measuring 1\text{ cm} \times 1\text{ cm} and has an area of 1\text{ cm}^2.

The image above shows that we can represent 4\text{ cm} \times 4\text{ cm} square as a region filled by sixteen 1\text{ cm} \times 1\text{ cm} squares. This means that the area of the square is simply equal to 16 \times 1 \text{cm}^2 = 16\text{cm}^2.  A faster way of counting the squares is to multiply the number of squares in each row and in each column. Extending this to find the area of a square, simply multiply the dimensions of the square. This leads to a faster way to find areas of squares given their sides’ lengths.

The Formula for the Area of the Square Using Its Side

Now that we’ve broken down the idea behind the area of a square, let’s establish a general formula for it. Since squares have four equal sides, a square’s length and width will always be equal. This means that when given the length of the square’s side, its area is equal to the product of the side’s length and itself or the square of the side’s length.

\text{Area of a Square} = \text{Side} \times \text{Side} \text{ unit}^2

\text{Area of a Square} = \text{Side} \times \text{Side} \text{ unit}^2

\begin{aligned}\text{Area of a Square} &= \text{Side} \times \text{Side} \text{ unit}^2\\ &= \text{Side}^2 \text{ unit}^2\\A &= s^2\\\end{aligned}
  • A = area of the square
  • s = length of the side
Square DimensionsArea of Square (A =s^2 )
2\text{ in}\times 2\text{ in} \begin{aligned}A &= 2^2 \text{ in}^2\&= 4 \text{ in}^2\end{aligned}
5\text{ cm}\times 5\text{ cm} \begin{aligned}A &= 5^2 \text{ in}^2\&= 25 \text{ cm}^2\end{aligned}
8\text{ km}\times 8\text{ km} \begin{aligned}A &= 8^2 \text{ km}^2\&= 64 \text{ km}^2\end{aligned}

The formula is easy to apply, so give it a try and work on the problems we’ve prepared for you to get used to the process. Let’s now see what happens if we’re given the square’s diagonal instead. This requires an understanding of radical numbers and the Pythagorean theorem, so if this hasn’t been discussed yet, head over to the next section to try out different examples!

The Formula for the Area of the Square Using Its Diagonal

When given the diagonal of a square, d, use the Pythagorean theorem to find the formula for the square’s area. Through special triangle, we can establish that d = \sqrt{2}s. Squaring this equation will lead to the formula for the square’s area in terms of d.

\begin{aligned}d^2 &= (\sqrt{2}s)^2\\d^2 &= 2s^2\\\dfrac{1}{2}d^2 &= s^2\\A&= \dfrac{d}{s^2}\end{aligned}

This simply means that we can find the area of the square by squaring its diagonal’s length and dividing the result by 2. In the next section, we’ll learn how to decide which approach would be best depending on the given dimensions and figures.

How To Find the Area of a Square?

Summarizing what we’ve learned so far, we can find the area of a square in three ways:

  • By counting the unit squares that make up the larger square then multiplying the number of squares by 1 squared unit.
\begin{aligned}\text{Area} &= \text{Number of Squares} \times 1 \text{ unit}^2\end{aligned}
  • When given the side of the square, square its length to find the area of the square.
\begin{aligned}\text{Area}&= \text{Side}^2 \text{ unit}^2\end{aligned}
  • If we have the length of the square’s diagonal, we can simply square it then take the half to find the square’s area.
\begin{aligned}\text{Area}&= \dfrac{1}{2}(\text{Diagonal})^2 \text{ unit}^2\end{aligned}

Let’s try out different examples to know the three methods by heart and work on different problems that involve the square’s area.

Problem 1

We begin with finding the area of the square shown below given that one-unit square has an area of 1\text{ inch}^2.

Count the total number of unit squares by multiplying the number of unit squares found in one row and one column. The square is a 6 \times 6 square, so it has six unit squares in its row and column. Its area is equal to 6 \times 6 = 36 times the area of one unit square.

\begin{aligned}A &= 36 \times 1\text{ in}^2\&= 36 \text{ in}^2\end{aligned}

This means that the square has an area of 36 squared inches or 36\text{ in}^2 .

Problem 2

Now, let’s try to find the area of the square floor that has a side of 40m. Since we’re given the measure of its side, we simply square its value to find the square’s area in squared meters.

\begin{aligned}A &= s^2\&= (40)^2 \text{ m}^2\&= 1600 \text{ m}^2\end{aligned}

This shows that the area of the square floor is equal to 1600 squared meters or 1600 m2 [/katex].

Problem 3

Let’s now try calculating the area of the square shown below. As we can see, what’s given now is the diagonal of the square, so we’ll use the third formula we’ve discussed to find its area.

\begin{aligned}A &= \dfrac{1}{2}d^2\&= \dfrac{1}{2}(12)^2 \text{ cm}^2\&= \dfrac{144}{2} \text{ cm}^2\&= 72 \text{ cm}^2\end{aligned}

Hence, the square’s area is equal to 72\text{ cm}^2 .

Problem 4

Now, let’s see what happens when we’re given the square’s perimeter instead. Suppose that a square field has a perimeter of 60 yards. What is the field’s area?

Recall that the perimeter of a square is equal to four times its side’s length, so the square’s side will be equal to the perimeter divided by four.

\begin{aligned}P &= 4s\\s&= \dfrac{P}{4} = \dfrac{60}{4} = 15\end{aligned}

The field has a side length of 15 yards, so we can now find the field’s area by squaring the length of its side.

\begin{aligned}A &=s^2\&= (15)^2 \text{ yards}^2\&= 225 \text{ yards}^2\end{aligned}

Hence, the square field has an area of 225 \text{ yards}^2.

The fourth problem is an example of how areas of squares and the corresponding formulas are used to solve word problems. When a surface is involved in a problem, there’s a high chance that areas are involved. In the next example, we’ll show you a problem involving square walls!

Problem 5

Allison plans to redesign her room and she begins by redoing her square walls’ wallpaper. Each wall has a side length of 8 \text{ feet} and it costs \$1.20 per squared feet for the new wallpaper. How much will it cost Allison to redo all four walls?

To find the cost for one wall, calculate the square wall’s area then multiply it with the cost it takes per squared foot. Since the wall has a side length of 8\text{ feet}, so take the square of the length to find its area.

\begin{aligned}A &=s^2\&= (8)^2 \text{ feet}^2\&= 64\text{ feet}^2\end{aligned}

Since it costs \$1.20 per squared foot for the new wallpaper, multiply the wall’s area by \$1.20.

\begin{aligned}\text{Cost} &= (\$ 1.20 /\text{ft}^2)(64\text{ ft}^2)\&= \$76.80\end{aligned}

It will cost Allison \$76.80 to redo the wallpaper for each wall, so multiply the cost by 4 to find the total cost for redoing the wallpaper for all four walls.

\begin{aligned}\text{Total Cost} &= 4(\$76.80)\&= \$307.20\end{aligned}

This means that the total cost to redo the wallpaper is \$307.20.