 The geometrical properties of a square or Perimeter and the area of Square both are the same. A square is a quadrilateral with has four equal sides. There are many square objects around you. Each square shape is characterized by one dimension, which is side length. In this chapter, we will learn about the geometrical properties of a square it means how to find the perimeter and area of a square. A rectangle is also called a square only if all its four sides are equal in length.

Square – Definition

Square will be a regular quadrilateral, which has all four sides have equal length and the four angles also are equal. The angles of the square are at right-angles or 90 degrees. A square is also defined as a rectangle whose two opposite sides have equal length. The diagonal of the square are equal and bisect each other at 90 degrees.

A Square may be a four-sided polygon that has all sides equal and the measure of the angles are 90 degrees. The shape of the square is, if we cut by a plane from the middle, then both the halves are symmetrical. Each half of the square seems like a rectangle with opposite sides equal. The above figure represents a square, all sides are equal and each angle equal to 90 degrees.

Properties of a Square:
A few important properties of a square are listed below:

1. All four interior angles are equal to 90 degrees.
2. The opposite sides of a square are parallel to each other.
3. The square has 4 vertices and sides.
4. All four sides of a square are equal to each other.
5. The diagonals of the square bisect one another at 90 degrees.
6. The length of the diagonals is greater than that of the sides of the square.

Geometrical Properties of Square In the above figure square ABCD,
AB = BC = CD = DA
AC = BD
∠ABC = ∠BCD= ∠CDA = ∠DAB= 90°.
AB and BD will be perpendicular bisectors of each other.
The diagonals are bisect each other at right angles.
OC = OA, OD=OB.
So, the Area of the ∆AOB = Area of the ∆BOC = Area of the ∆COD = Area of the ∆DOA.

Perimeter and Area of Square

The two main properties that outline a square, the properties are Area and Perimeter. Let discuss them one by one:

The perimeter of a Square: Any shape that will be laid on a flat surface is called a two-dimensional object. The length of the side or boundary of any two-dimensional shape is called the perimeter. The perimeter of a square is the sum of all four sides length of a square. Square is one of the 2D shapes having four equal sides and four corners angles are 90 degrees each. The units of the perimeter remain the same as that of the side-length of the square.

So, the perimeter of a square is, a sum of all the sides that is,
Perimeter = Side + Side + Side + Side = 4a.
Let ‘a’ is the side of a square. So, it will be 4a.

Area of a Square: The area of a square is the region covered by it during a two- dimensional plane. The area of a square is adequate to the edges or side squared. It is measured in square units.
So, the Area of a square is side x side
The area of a square is a² sq. unit.
If ‘a’ is that the length of the side of the square.

Length of Diagonal of Square

The length for the diagonals of the square is equal to s√2, where s is the side for the square. As the length for the diagonals is equal to one another, so the diagonal is the hypotenuse and the two sides of the triangle formed by the diagonal of the square, are perpendicular and base.
Since, Hypotenuse² = Base² + Perpendicular²
Hence, Diagonal² = side² + side²
Diagonal = √2side²
d = s√2
Where ‘d ‘ is the length of the diagonal of a square and s is that the side of the square.

Geometric Properties of Square Examples with Solutions

Problem 1: What is the perimeter of a square and also find the cost of fencing a square park of side 150 metres at the rate of Rs 30 per metre.

Solution:
As given in the question, the data.
To find the cost of fencing a square park we need to find out the length of the boundary of a square park.
Now, we will find out the perimeter of a square.
The perimeter of a square = 4a
The side of the square park is 150 metres then,
The perimeter of a square = 4 x 150 = 600 metres.
Next, the cost of fencing 1 metre is Rs 30.
So, the cost of fencing 600 metres will be 600 x 30  i.e, Rs.18,000
Therefore the cost of fencing a square park of sides 150 metres is Rs 18,000.

Problem 2: If the perimeter of a square is 40m, then what is the side of the square?

Solution:
Given in the question, the values are
The perimeter of a square is 40m.
Now, we will find the side of a square value.
We know that, Perimeter = 4xside
So, side = perimeter/4
Substitute the value in the above formula.
Side = 40/4cm = 10 cm.
Hence, the side of a square value is 10 cm.

Problem 3: A square has a side equal to 5cm. Find the area of a square, perimeter of a square and length of diagonal of a square.

Solution:
Given in the question, the values are
The Side of the Square, S = 5 cm.
Now, we will find the area, perimeter, and length of the diagonal.
We know that,
The Area of a square is, S² = 5² = 25 cm²
Next, the Perimeter of the square = 4xS= 4 × 5 cm = 20cm.
Length of the diagonal of square = S√2 = 5 × 1.414 = 7.07
Thus, the area, perimeter, and the length of diagonal of a square is 25 cm², 20cm, 7.07.

Problem 4: Find the area of a square, whose dimensions is 4m 15cm.

Solution:
As given in the question, the value is 4m 15cm.
We know that, 1m = 100cm.
So, 4m 15cm = 4m = 400 cm
400 cm + 15 cm = 415cm.
Now, we have to find out the area of a square.
The area of a square is a².
Substitute the value in the above formula.
Area of a square = (415 cm)²= 415 x 415 = 172,225 cm.
Thus, the area of a square is 172,225 sq. cm

FAQ’S on Geometrical Properties of Square

1. What is the formula for the perimeter of a square?

A. All the sides of a square are equal, we need one side to find its perimeter. The perimeter of the square is written as s+s+s+s = 4s
Therefore, the formula of the perimeter of a square is 4x(length of any one of the sides) = 4s.

2. Is square a polygon?

A. A square is a four-sided polygon, which has all its sides length is equal. It is also a type of Quadrilateral.

3. What are the examples of Square?

A. There are many examples of square shape in real-life such as a square plot or field, a square-shaped ground, square-shaped table cloth, the tiles of the floor in a square shape, etc.

4. What are the five properties of a square?

A. The following are the five important properties of a square:

• All the four interior angles of a square are equal to 90°.
• The opposite sides of the square are parallel to each other.
• The two diagonals of the square are equal to each other.
• The diagonal of the square is divided into two similar isosceles triangles.
• The length of the diagonals is greater than the sides of the square.