Difference between a Square and a Rhombus (Square Vs Rhombus) – BYJUS

Difference between a Square and a Rhombus

A polygon having four sides is known as a quadrilateral. A quadrilateral can further be classified as a rectangle, square, rhombus, trapezoid, kite, or parallelogram according to their special properties. They have some properties in common and some differences. In this article, we will focus on the difference between a square and a rhombus....Read MoreRead Less

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What is a Square?

A square is a parallelogram that is equilateral and equiangular. The measure of each angle of a square is always 90°. The diagonals of a square are equal in length and they bisect each other at right angles.

 

Square

Properties of a Square

  • The opposite sides are parallel to each other.
  • Diagonals are perpendicular bisectors of each other.
  • The sides are perpendicular to each other.
  • The diagonal of a square is equal to \(\sqrt {2}\) times its side.
  • The diagonal of a square divides it into four congruent triangles of equal areas.
  • If the side of a square is ‘a’, its perimeter is equal to 4a.
  • If the side of a square is ‘a’, its area is equal to a\(^2\).

What is a Rhombus?

A rhombus is a parallelogram that is equilateral but not always equiangular. The diagonals of a rhombus are not equal but bisect each other at a right angle.

rhombus1

The Properties of a Rhombus

  • The opposite sides are parallel to each other.
  • The diagonals are perpendicular bisectors of each other.
  • The sides are equal in length
  • A diagonal bisects a pair of opposite angles.
  • If the side of a rhombus is ‘a’, its perimeter is equal to 4a.
  • If the diagonals of a rhombus are \(d_1\) and \(d_2\) respectively, its area is: \(\frac{1}{2}\times d_1 \times d_2\)

Differences between a Square and a Rhombus

Square

Rhombus

All the angles of the square are equal and they are 90°. 

Opposite angles of the rhombus are equal but need not be a right angle.

Diagonals are equal and equal to \(\sqrt{2}\) times of the side of the square.

Diagonals are not of equal length in a rhombus.

All squares are rhombuses.

All rhombuses cannot be squares, because rhombuses are not always equiangular.

The adjacent sides of the square are perpendicular to each other.

The adjacent sides of the rhombus are not always perpendicular to each other. 

The measure of an angle of a square bisected by a diagonal is 45°.

In a rhombus, the diagonals bisect the angles, but they need not always be 45°.

If the side of a square is ‘a’, its area is equal to a\(^2\).

If the diagonals of a rhombus are \(d_1\) and \(d_2\), respectively, its area is \(\frac{1}{2} \times d_1 \times d_2\)

We can inscribe a square in a circle.

We can not inscribe a rhombus in a circle.

A square is always symmetrical about four lines.

A rhombus is always symmetrical about two lines. 

Similarities between a Square and a Rhombus

Square

Rhombus

A square is a parallelogram. 

A rhombus is also a parallelogram.

The measures of the sides of a square are equal.

The measures of the sides of a rhombus are equal.

In a square, the opposite sides are parallel to each other.

In a rhombus, the opposite sides are also parallel to each other.

The diagonals of a square are perpendicular bisectors.

The diagonals of the rhombus are also perpendicular bisectors.

The sum of all interior and exterior angles is 360°.

The sum of all interior and exterior angles is 360°.

If the side of a square is ‘a’, its perimeter will be ‘4a’.

If the side of the rhombus is ‘a’, its perimeter will be 4a.

Solved Examples

Example 1: Identify whether the parallelogram ABCD is square or rhombus.

square_eg1

Solution:

Given that, the parallelogram ABCD is equilateral. 

So, it can be a square or a rhombus. 

If the diagonals are of equal length, it is a

square, otherwise it is a rhombus.

We know that the diagonals of a parallelogram bisect each other.

So, AC = AO + OC

= 5 + 5

= 10 cm

And, BD = BO + OD

= 7 + 7

= 14 cm

We can see that, AC \(\neq\) BD

Since AC \(\neq\) BD, ABCD is not a square and it is a rhombus.

 

Example 2: Check whether the given parallelogram ABCD with vertices A(2,0), B(7,0), C(7,5), and D(2,5) is a square or not.

 

Square

 

 

Solution :

Find all the lengths of the sides of the parallelogram.

AB = \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) Distance formula

= \(\sqrt{(7-2)^2+(0-0)^2}\)              Substitute

= \(\sqrt{5^2}\)                                         Evaluate square root

 AB = 5

BC = \(\sqrt{(x_3-x_2)^2+(y_3-y_2)^2}\) Distance formula

= \(\sqrt{(7-7)^2+(5-0)^2}\)              Substitute

= \(\sqrt{5^2}\)                                         Evaluate square root

BC = 5

CD = \(\sqrt{(x_4-x_3)^2+(y_4-y_3)^2}\) Distance formula

= \(\sqrt{(2-7)^2+(5-5)^2}\)              Substitute

= \(\sqrt{(-5)^2}\)                                   Evaluate square root

 CD = 5

                          

 DA = \(\sqrt{(x_1-x_4)^2+(y_1-y_4)^2}\) Distance formula

= \(\sqrt{(2-2)^2+(0-5)^2}\)              Substitute

= \(\sqrt{(-5)^2}\)                                   Evaluate square root

 DA = 5 

                         

AC = \(\sqrt{(x_3-x_1)^2+(y_3-y_1)^2}\) Distance formula

= \(\sqrt{(7-2)^2+(5-0)^2}\)              Substitute

= \(\sqrt{5^2+5^2}\)                                 Evaluate square root

AC = \(5\sqrt{2}\)

                        

BD = \(\sqrt{(x_4-x_2)^2+(y_4-y_2)^2}\) Distance formula

= \(\sqrt{(2-7)^2+(5-0)^2}\)              Substitute

= \(\sqrt{(-5)^2+5^2}\)                           Evaluate square root

BD = \(5\sqrt{2}\)

Side AB = BC = CD = DA

and diagonals BD = AC

Hence, we can say that ABCD is a square.

Frequently Asked Questions

A parallelogram is a quadrilateral whose opposite sides are parallel.

The key difference between a square and a rectangle is that all the sides of a square are equal in length, whereas in a rectangle, only the opposite sides are equal in length.

A quadrilateral is a closed figure with four sides. So, a square is a quadrilateral.

If the angles of a rhombus are right angles, we can say that the rhombus is a square.