**Properties Of Square:** Looking for Properties of Square? You have come to the right place. This article will provide you with all the necessary details about Square – definition, diagram, square properties, and formulas, along with examples.

Square is a plane geometrical figure. Knowledge of various concepts related to Square is extremely important to solve various problems of geometry and mensuration. So, read this article and get a clear understanding of Properties of Square.

## Properties Of Square

Questions related to Square 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 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 are easier compared to the rest of quadrilaterals.

### Properties Of Square

Some of the basic properties of Square are as under:

a. A square is a quadrilateral with four equal sides and four equal internal angles.

b. It is a rhombus with four equal angles (each angle equals to 90°).

c. A square is a rectangle with its two adjacent sides equal.

d. It is a parallelogram with all four internal angles equal to 90° and adjacent sides equal in length.

e. The opposite sides of a square are parallel to each other.

f. The diagonals of a square are equal in length, bisect each other, and are perpendicular to each other.

g. Each diagonal of a square divides the square into two equal, isosceles triangles.

h. The diagonal of a square bisect the internal angles at the two points joining it.

i. 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 center at the point of bisection and its circumference passing through the four vertices of the square.

j. Similarly, an incircle can be formed with its center at the point of bisection and its circumference touching the sides of the square.

k. The diagonals of the square are diameters of the circumcircle.

l. The radius of the incircle is equal to half of the side of the square.

### Formulas Of Square

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

i. Area of a Square, A = *a*^{2}, where *a* is the length of each side of the square.

Area of a Square = (Side)^{2} = a^{2} |

ii. Perimeter of a Square, S = Total length of all sides of the square =* 4a*.

Perimeter of a Square = 4 × Side = 4a |

iii. 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, *r*1 = 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, *r*2 = 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 = *a*^{2} = *6*^{2} = 36 cm^{2}

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 = a^{2} = (3.54)^{2} =12.5 cm^{2}

So, now you know the Properties of Square. Solve more number of questions and master the topic.

### Other Articles On Maths

Some other helpful Maths articles by Embibe are provided below:

HCF And LCM |

Properties Of Triangles |

BODMAS Rule |

Properties Of Circle |

Pythagoras Theorem |

Properties Of Rectangle |

These simple concepts lay the foundation required to understand complex concepts in higher classes. Also, these will be of tremendous help to you in cracking the Numerical Ability/Quantitative Aptitude section of various competitive exams.

At Embibe, you can solve free Maths practice questions for Class 8, 9, 10, 11, and 12 and get detailed solutions and real-time feedback:

a. Class 8 Maths Practice Questions |

b. Class 9 Maths Practice Questions |

c. Class 10 Maths Practice Questions |

d. Class 11-12 Maths Practice Questions |

So, make the best use of these resources and master the subject.

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

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