Let’s explore several methods to find the area of a square: using the sides, diagonal, perimeter, and if it is inscribed in a circle. By using the simple formulas discussed in this article, you won’t have to spend extra time calculating the area of this geometric figure.
But before we move on to the topic of how to find the area of a square, let’s recall what this figure is and its characteristics. A square is a quadrilateral where all sides are equal and each angle is 90 degrees.
The area of a square is the surface it occupies on a plane. It is calculated in cm², dm², m², etc., and is denoted by the letter S.
How to Find the Area of a Square Using Geometric Formulas
In middle school math classes, children learn various ways to find the area of a square. The simplest way is to calculate the surface of the figure using the length of its sides. For this, you only need to multiply them together. The resulting number will be the desired result.
We have already mentioned that in any square figure, all sides are of equal length. If we denote them by the letter a, the formula for finding S in this case will look like this: S = a * a, or S = a².
Let’s consider an example of how to find the area of a square using this formula. Imagine we have a square with a side of 4 cm. Since all its sides are of equal length, to calculate S□ we perform the following actions: Multiply the sides: S = 4 * 4 = 4². Perform the calculation and get S = 16.
Answer: 16 cm².
Earlier wa wrote: How to Find the Perimeter of a Rectangle – Formula, 5 Example Problems

Using the Diagonal
When the length of a square’s side is unknown, there are other ways to calculate its S. This can be done using the diagonal d.
To calculate the area in this case, you need to square the length of the diagonal and divide it by 2. In the form of a formula, this can be written as: S = d²/2.
Let’s consider the use of this formula in practice. Suppose we have a flat □ with a diagonal d equal to 12 cm. We need to find its S. Perform the following calculations:
Substitute the known data into the formula: S = 12²/2. Perform the mathematical calculations: S = 144/2 = 72.
Answer: S = 72 cm².

Using the Perimeter
In some problems, the length of the diagonal or side of the figure is not given, but the perimeter is known. In this case, there is no other option but to find S□ using the perimeter.
First, let’s recall that the perimeter is the sum of the lengths of all the sides of the figure. Since all sides of a square are equal, it is determined by the formula P = 4a, where: P — perimeter; a — side length.
Using this formula, you can determine the length of the side of the geometric figure: a = P/4.
After this, to calculate the necessary value, use the already known formula S = a².
Let’s see how to find the area of a square in this way using an example of a figure with a perimeter of 36 cm. To solve the problem, perform the following steps: Calculate the length of each side: a = 36/4 = 9 cm. Calculate the area: S = 9² = 81 cm².
So, the answer is 81 cm².

If Inscribed in a Circle
The diagonal of a square is equal to the diagonal of the circumscribed circle. Since this figure is a regular rhombus, its area can be found using the formula for calculating the S of a rhombus, which is half the product of its diagonals.
The square has equal diagonals, so the calculation formula looks as follows: S = d²/2.
Now let’s calculate the S□ inscribed in a circle, knowing that the diagonal of the circle is 10 cm. Solution: Since the diagonals of the circle and the figure inscribed in it are equal, use the formula for calculating S using the diagonal: S = d²/2 = 10²/2. Perform the calculations: S = 100/2 = 50 cm².
Answer: 50 cm².

Video on How to Find the Area of a Square
For a more in-depth study of the topic, we suggest you watch a YouTube video that explains S□ and provides examples of problem-solving.
Conclusion
The area of a square can be found in several ways, depending on what information is known: the length of the sides, the diagonal, the perimeter, or the radius of the circumscribed circle. Using these formulas allows you to quickly and accurately calculate the area of a figure for various practical tasks. Knowing these methods is useful in many fields, from construction to design.
Frequently Asked Questions About How to Find the Square Area
It is the size of the surface occupied by a square shape on a plane.
According to the international measurement system, it is calculated in square centimeters, meters, kilometers, etc.
For this, you need to multiply its length by its width. Since the sides of the figure are equal, the formula used is S = a².
Yes, for this, you need to divide its diagonal, squared, by two. The resulting number will be the desired result.
The ability to find it can be useful in many situations: during architectural planning, construction, and repair work, in computer graphics, design, landscape planning.