Square Formula

Square is a regularÂ quadrilateral. All the four sides and angles of a square are equal. The four angles are 90 degrees each, that is, right angles. A square may also be considered as a special case of rectangle wherein the two adjacent sides are of equal length.

In this section, we will learn about the square formulas â€“ a list of the formula related to squares which will help you compute its area, perimeter, and length of its diagonals. They are enlisted below:

$$\begin{array}{l}Area\; of\; a\; Square\; = a^{2}\end{array}$$

Perimeter of a Square =4a

$$\begin{array}{l}Diagonal\; of\; a\; Square\; = a\sqrt{2}\end{array}$$

Where â€˜aâ€™ is the length of a side of the square.

Properties of a Square

• The lengths of all its four sides are equal.
• The measurements of all its four angles are equal.
• The two diagonals bisect each other at right angles, that is, 90Â°.
• The opposite sides of a square are both parallel and equal in length.
• The lengths of diagonals of a square are equal.

Derivations:

Consider a square with the lengths of its side and diagonal are a and d units respectively.

Formula for areaÂ of a square: AreaÂ of a square can be defined as the region which is enclosed within its boundary. As we mentioned, a square is nothing a rectangle with its two adjacent sides being equal in length. Hence, we express area as:

The area of a rectangle = Length Ã— Breadth

Here,

$$\begin{array}{l}Area\; of\; square\; = a \times a = a^{2}\end{array}$$

The formula for the perimeterÂ of a square: Perimeter of the square is the length of its boundary. The sum of the length of all sides of a square represents its boundary. Hence, the formula can be given by:

Perimeter = length of 4 sides

Perimeter = a+ a + a + a

Perimeter of square = 4a

The formula for diagonalÂ of a square: A diagonal is a line which joins two opposite sides in a polygon. For calculating the length diagonal of a square, we make use of the Pythagoras Theorem.

In the above figure, the diagonalâ€™ divides the square into two right angled triangles. It can be noted here that since the adjacent sides of a square are equal in length, the right angled triangle is also isosceles with each of its sides being of length â€˜aâ€™.

Hence, we can conveniently apply the Pythagorean theorem on these triangles with base and perpendicular being â€˜aâ€™ units and hypotenuse beingâ€™ units. So we have:

$$\begin{array}{l}d^{2}=a^{2}+a^{2}\end{array}$$

Or

$$\begin{array}{l}d=\sqrt{2a^{2}}=a\sqrt{2}\;units\end{array}$$

Solved examples:

Question 1: A square has one of its sides measuring 23 cm. Calculate its area, perimeter, and length of its diagonal.

Solution: Given,

Side of the square = 23 cm

Area of the square:

Area of the square =

$$\begin{array}{l}a^{2}=23^{2}=529 cm^{2}\end{array}$$

Perimeter of the square:

Perimeter of the square= 4a= 4 Ã— 23 = 92 cm

Diagonal of a square:

Diagonal of a square =

$$\begin{array}{l}a\sqrt{2}=23\sqrt{2\;cm}=32.52cm\end{array}$$

Question 2: Â A rectangular floor is 50 m long and 20 m wide. Square tiles, each of 5 m side length, are to be used to cover the floor. Find the total number of tiles which will be required to cover the floor.

Solution: Given,

Length of the floor = 50 m

$$\begin{array}{l}No.\; of\; tiles\; needed = \frac{Area\; of\; floor}{Area\; of\; one\; tile}=\frac{1000}{25}=40\; tiles\end{array}$$