Properties of Square: The properties of squares is one of the most important concepts in Mathematics. Students should be well-versed with the formulas and properties of square numbers to have a good command over Maths. Without understanding the concept it is hard to move forward and learn more complex concepts in Mathematics. The properties of squares find application in various theorems and functions, algebraic expressions, algebraic identities and more.

This article has provided in-depth explanation on properties of square numbers and other important details related to it for students. After reading this article students will be able to improve their grades and ace their examination. To know more, read the article till the end.

All Properties of Square

Questions related to properties of squares and rectangles are asked not only in school-level exams but also in various competitive exams for job recruitment as well as in MBA entrance exams like CAT and MAT.

So, let us start off with the definition of a Square.

What is a Square?

A square is a two-dimensional, closed geometrical shape that has four sides, four vertices, and four angles. All four sides of a square are equal in length. So are the four internal angles of a square, each of which measures 90°.

Basically, a square is a rectangle whose adjacent sides are equal in length.

A square is a quadrilateral. Some other types of quadrilaterals are:

a. Rectangle
b. Parallelogram
c. Kite
d. Rhombus
e. Trapezoid/Trapezium

Solving questions related to squares is easier compared to the rest of the quadrilaterals.

Properties of Diagonals of Square

Some of the basic properties of Square are as under:

• A square is a quadrilateral with four equal sides and four equal internal angles.
• It is a rhombus with four equal angles (each angle equals 90°).
• A square is a rectangle with its two adjacent sides equal.
• It is a parallelogram with all four internal angles equal to 90° and adjacent sides equal in length.
• The opposite sides of a square are parallel to each other.
• The diagonals of a square are equal in length, bisect each other, and are perpendicular to each other.
• Each diagonal of a square divides the square into two equal, isosceles triangles.
• The diagonal of a square bisects the internal angles at the two points joining it.
• The two diagonals of a square bisect each other. So, the four vertices of the square are equidistant from the point of bisection. This means, a circle can be formed with its centre at the point of bisection and its circumference passing through the four vertices of the square.
• Similarly, an incircle can be formed with its centre at the point of bisection and its circumference touching the sides of the square.
• The diagonals of the square are diameters of the circumcircle.
• The radius of the incircle is equal to half of the side of the square.

Check other important Maths articles:

Algebra Formulas	Log Table
Geometry Formulas	Probability Formula
Arithmetic Progression Formulas	Compound Interest Formula
Trigonometry Formulas	Permutation and Combination
Mensuration Formulas	Differentiation Formulas
HCF and LCM	BODMAS Rule

Formulas of Square

Some of the important formulas related to Square are as under:

i) Area of a Square, A = a², where a is the length of each side of the square.

Area of a Square = (Side)2 = a2

ii) Perimeter of a Square, P = Total length of all sides of the square = 4a.

Perimeter of a Square = 4 × Side = 4a

iii) Diagonal of a Square, D: Each of the diagonals of a square divides the square into two right-angled triangles with the diagonal of the square being the hypotenuse of the triangles.

Now, applying Pythagoras’ Theorem which states that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides, we have:

(Diagonal)² = (Side)² + (Side)².

For a square, both sides are equal in length. So, we have:

(Diagonal)² = 2(Side)²
Or Diagonal, D = √2 × Side

From this,

Length of Diagonal (D) = √2 × Side = a√2

iv) Radius of the Circumcircle of a Square, r1 = Half of the Diagonal of the Square = a√2/2 = a/√2.

Radius of the Circumcircle of a Square = (1/2) × Diagonal (D) = a/√2

v) Radius of the Incircle of a Square, r2 = Half of the Side of the Square = a/2

Radius of the Incircle of a Square = (1/2) × (Side) = a/2

Problems on Square

Here are a few solved problems on Square:

Example 1: Find the area, perimeter, and length of diagonal of a square of side 6 cm.

Solution: Here, length of each side of the square, a = 6 cm.
Therefore,
Area = a²
= 6²
= 36 cm²
Perimeter = 4a
= 4 × 6
= 24 cm
Diagonal = a√2
= √2 × 6
= 8.485 cm.

Example 2: If the diagonal of a square measures 5 cm, find its area.

Solution: We know that the diagonal, D of a square of side a is given by:
D = a√2
Here, D = 5 cm.
Therefore, we have:
5 = a√2
⇒ a = 5/√2
= 3.54 cm
Now, area of the square = a²
= (3.54)²
=12.5 cm²

Also check,

Properties of Rhombus	Properties of Quadrilaterals
Properties of Parallelogram	Properties Of Triangles
Properties of Rectangle	Properties of Circle

Frequently Asked Questions on All Properties of Square

Here we have provided some of the frequently asked questions:

Q1: What are the properties of Square?
Ans: A Square has the following properties:
(i) All four interior angles are equal and each measures 90°.
(ii) All four sides of the square are congruent or equal to each other.
(iii) A square is a rectangle with two adjacent sides equal.
(iv) The opposite sides of a square are parallel to each other.
(v) The diagonals of a square are equal in length, bisect each other, and are perpendicular to each other.

Q2: Is a Square a Rhombus?
Ans: No, a Square is different from Rhombus in the way that a Square has all its angles equal to 90°, but a rhombus only has its opposite angles equal.

Q3: What is the difference between a Square and a Rectangle?
Ans: A Square has all sides equal and opposite sides parallel, whereas, for a rectangle, only the opposite sides are equal.

Q4: What are some of the real-world examples of a Square?
Ans: Some of the real-life examples of a Square are Carrom Board, Square tiles, Square-shaped table, etc.

Q5: Is a Square a type of quadrilateral?
Ans: Yes, a Square is a type of quadrilateral whose all four sides are equal and each angle measures 90°.

Q6. How to find the length of the diameter of a square if the side length is given?
Ans: The formula to calculate the diameter of a square is a√2 where a is the side length of the square.

Q7. What is the area of a square?
Ans: The area of a square is given by the square of its sides.

So, now you know the Properties of Square. Solve more questions and master the topic. At Embibe, you can solve Maths practice questions and mock tests for Classes 8, 9, 10, 11, and 12 and get detailed solutions and real-time feedback. So, make the best use of these resources and master the subject.

We hope this article on the Properties of Square helps you. If you have any queries, feel to ask in the comment section below. We will get back to you at the earliest.