Square (Geometry) – Definition & Properties

What is Square?
Square is a quadrilateral of four sides in which all sides are equal, opposite sides are parallel and all angles measures 90 degrees.

The important terms for square are:
⟹ Equal sides
⟹ All angles measures 90 degree

Structure of Square

What is a square

Given above is the image of square.

Note that:
(a) The square has 4 equal sides
AB = BC = CD = DA

(b) It has 4 vertex
Point A, B, C & D are the vertex

(c) Square has 4 angles each measuring 90 degrees
∠A =∠B = ∠C = ∠D = 90 degrees

Examples of Square

examples of square

Given above are the examples of square.

The image (ii) & (iii) may appear different but they are squares, I have just tilted it to make it appear different.

No matter how the above figures look, they all have same property:
(a) All sides equal
(b) Each angle is 90 degrees

Properties of Square

(01) All interior angle exactly measures 90 degree

angles of square

There are four angles in a square and all measure 90 degree.
∠A = ∠B = ∠C = ∠D = 90 degree

Also, If you add all the angles you will get 360 degree
⟹ ∠A + ∠B + ∠C + ∠D
⟹ 90 + 90 + 90 + 90
⟹ 360 degree

(02) In a square all sides are equal

All sides of square are equal

Given above is the figure of square ABCD
So, AB = BC = CD = DA = 2 cm

(03) Two diagonals of square are equal to each other

diagonals of square

Given above are the diagonals AC & BD of a square.

According to diagonal property;
AC = BD (both diagonals are equal)

(04) Formula for Diagonal length

If the length of square is a cm, then formula for length of diagonal is:

Diagonal Length of square


Proof

Diagonals of square are equal

Given above is square ABCD with length of side a cm and diagonal BD

To Prove
Length of BD = \sqrt{2} \ a

Solution
We know that all angles of square measures 90 degree.

Applying Pythagoras Theorem on triangle BCD, we get

BD^{2} =\ CD^{2} +\ CB^{2}\\\ \\ BD^{2} =\ a^{2} +\ a^{2}\\\ \\ BD^{2} =\ 2\ a^{2}\\\ \\ BD\ =\sqrt{2} \ a

Hence proved

(05) Diagonals of square bisect each other at 90 degree angle

Diagonals of square intersect at 90 degree

Given the square ABCD with AC and BD as diagonals.

According to the square property:
(a) Diagonals bisect each other
AO = OC
DO = OB

(b) Diagonals AC & BD intersect at 90 degree angle

(06) Perimeter of Square

Perimeter is calculated by finding the length of complete boundary.

The formula for Perimeter of Square is:
Perimeter of square = Side + Side + Side + Side
Perimeter = 4 x Side


Example
If the side of square is 3 cm
Then, Perimeter of Square
4 x 3 cm
12 cm

(07) Area of Square Formula

The region covered by any figure is known as Area

The formula for Area of Square is:
Area = Length x Breadth

Since in square all sides are same, the formula becomes:
Area\ =\ Side\ \times \ Side\\ \\ Area\ =\ Side^{2}


Example
Below is the square of side 2 cm. Find its area.

Formula for Area of square

Area of Square = 2 x 2 = 4 sq. cm

(08) Symmetry for Square

A line which divides the figure into two equal halves is known as Line of Symmetry.

In square, there are four lines of symmetry.

Line of symmetry of Square

(09) Square’s Circumcircle

Square’s Circumcircle is a circle which passes through all the vertex of the square.

Circumcircle of square and its formula

Given above is square ABCD of side a cm

Circle with center O passes through all the vertex of the square, the circle is said to be circumscribing the square.


Radius of Circumcenter Formula
Note the center O is at the intersection of both diagonals AC & BD.

And we know that in square, diagonals bisect each other.

Diagonal\ of\ Square\ BD\ =\sqrt{2} \ a\\\ \\ Radius\ OB\ =\ \frac{\sqrt{2} \ a}{2}\\\ \\ Radius\ OB\ =\ \frac{a}{\sqrt{2}}

Radius of circumcircle of square

(10) Square’s Incircle

What is incircle?
It’s a circle which is present inside the square and touches all its sides

Incircle of square

Above image shows square ABCD with circle inside the figure.

You can easily observe that the radius of incenter is half of square’s side.

Hence, Incircle Radius = a/2

Formula for radius of incircle of  square

Frequently Asked Question – Square

(01) How us Square and Parallelogram different?

Read Solution

All Squares are parallelogram but all parallelogram are not squares.

Parallelogram has following property
(a) Opposite sides are equal and parallel
(b) Opposite angles are equal

All squares follow the above property, making a square a form of parallelogram

Difference between square and parallelogram

Above image is of parallelogram.
You can notice that it is not same as square.

(02) How square and rectangle are different?

Read Solution

Square have all equal sides.
While Rectangle have opposite sides equal

Difference between square and rectangle

(03) Both Square and Rhombus have all equal sides, so how they are different?

Square and Rhombus difference
Read Solution

Between Square and Rhombus, the difference lies in the fact that in Rhombus, the angles may or may not be 90.

Sometime students get confused with the properties of Rhombus.
Just remember one trick sentence for Rhombus:

” Rhombus is a Square with a twist”

It means that in rhombus, all the sides are equal ( like square) but its angles are twisted to give it distinct characteristic.

(04) Can we divide the square into two equal triangles?

Read Solution

YES!!
By joining one of the diagonals, the square can be divided into two equal triangles

Diagonal of Square

Given above is the image of square ABCD
By joining the diagonal BD, we get two equal triangles ABD & DCB

(05) Can square be found in real life?

Read Solution
Squares in Real life

Summary of Square Properties

Summary of Square Properties

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