Square Area

What is a Square?

A square is a polygon with four equal sides and right angles. All its sides are congruent (have the same length), and its diagonals are congruent (also have the same length). The points of intersection are called vertices, and from them, four internal angles of 90° each are formed.

How to Calculate the Area of a Square

The area of a square represents the number of surface units inside the square. To calculate this surface, you must first know the length of its side (a), that is, the length of the line connecting two vertices of the square.

Formula to Calculate the Area of a Square:

Square Area = side²

Practical Example

Let's say we need to find the area of a square, and we are given the value of its side (a), which, for this example, will be 4 [cm].

Replacing the value of the side in the area formula

If we already have the length of the side (4 cm), all that remains is to replace this value in the formula to calculate the area of the square. In effect:

Square Area = side²
Square Area = (4)²
Square Area = 16 [cm²]

Finding the Square Area from its Diagonal

It is possible that in a problem statement, the only data you have is the length or length of the diagonal of a square. In these cases, you can use the Pythagorean Theorem to determine the value of the square's side if you consider that the diagonal is the diagonal of an isosceles triangle with interior angles of 45°, 45°, and 90°, respectively.

Pythagorean Theorem Formula

Hypotenuse² = base² + height²

By definition, all sides or edges of a square are equal. It is valid to assume that for any triangle formed by the diagonal of a square, the base and height will be equal. If we consider the diagonal as the hypotenuse, we can rewrite the Pythagorean Theorem as follows:

Diagonal² = side² + side²
Diagonal² = 2 · side²
Diagonal = √2 · side

Solving for the side in the equation, we get the following relationship:

Side = Diagonal ÷ √2

This relationship holds for all squares and allows you to find the side length simply using any calculator. Once you know the side length, you can replace it in the formula to find the square's area mentioned in the yellow box above.

How to Calculate the Square Area from its Perimeter

Calculating the area of a square from its perimeter is very simple, since by definition, a square has 4 sides (a edges) of the same size. With this background, we can deduce that the side of a square is equivalent to its perimeter divided by 4.

Replacing in the square area formula, we have:

Square Area = side²
Square Area = (perimeter ÷ 4)²

Practical Example

Let's say we need to find the area of a square with a perimeter of 24 [cm]. Replacing the value of the perimeter in the area formula, we have:

Square Area = side²
Square Area = (perimeter ÷ 4)²
Square Area = (24 ÷ 4)²
Square Area = (6)² = 36 [cm²]