Rhombus

Almost every one of us knows that Rhombus is a quadrilateral, and therefore, just like other quadrilaterals such as square, rectangle, etc., has four vertices and four edges enclosing four angles. Nevertheless, this is not all. There's much more to know about this amazing 2D shape that acts as a crucial part of mathematics, one of the core subjects that walk with us from school to higher education. So, let's get familiar with all the crucial aspects of Rhombus.


Rhombus Definition

In Euclidean geometry, a rhombus is a special type of quadrilateral that appears as a parallelogram whose diagonals intersect each other at right angles, i.e., 90 degrees. As the shape of a rhombus is just like that of a diamond, it is also known as diamond. The diamond-shaped figure in the playing cards is one of the best examples of a rhombus. Moreover, possibly all the rhombi are kites and parallelograms, but if all angles of a rhombus measure 90°, then it is a square.  

 

In other words, a rhombus is a special type of parallelogram in which opposite sides are parallel, and the opposite angles are equal. Besides having four sides of equal length, a rhombus holds diagonals that bisect each other at 90 degrees, i.e., right angles. 


The below figure shows a rhombus ABCD with four equal sides - AB, BC, CD, and AD, and two diagonals AC and BD that intersect each other at right angles. 

                  

Now, after observing the above-given figure and reading the information mentioned above, you might be a bit confused about whether rhombus is square and vice-versa or not. If yes, then don't worry as here's the relevant solution for you.


Is Square a Rhombus?

A square is a quadrilateral with all its sides equal in length, and so do the rhombus. Also, the two diagonals of the square are perpendicular to each other and bisect opposite angles, just like the rhombus. Hence, it is clear that a square is a rhombus. On the other hand, as the basic property of square states that all its interior angles are right angles, a rhombus is not considered as square, unless all the interior angles measure 90°. 


Angles of Rhombus

Any rhombus includes four angles, out of which the opposite ones are equal to each other. Moreover, the rhombus consists of diagonals that bisect each other at right angles. In other words, we can say that each diagonal of a rhombus cuts the other into two equal parts, and the angle formed at their crossing points measure 90°.  The diagonals also bisect the opposite angles of the rhombus. 


Rhombus Formulas

Formulas for any rhombus are defined while concerning the two main attributes like area and perimeter.


Area of Rhombus

The area of a rhombus refers to the region covered by it in a 2D plane. Based on this definition, the formula for the area of a rhombus is equal to the product of its diagonals divided by 2, and can be represented as:


Area of Rhombus (A) = (d1 x d2)/2 square units, where d1 and d2 are the diagonals of the rhombus


Perimeter of Rhombus

The perimeter of a rhombus is defined as either the total length of its boundaries or the sum of all the four sides of it. Hence, the formula for the perimeter of a rhombus can be represented as:

 

The perimeter of Rhombus (P) = 4a units, where ‘a’ is the side of the rhombus. 


Properties of Rhombus

Now, have a look at some of the significant properties of the rhombus. 


  • All four sides are equal in length

  • Opposite sides are parallel

  • Opposite angles are equal 

  • Diagonals bisect each other at right angles, i.e., 90 degrees

  • Rhombus's diagonals bisect its opposite angles 

  • The sum of two adjacent angles is supplementary, i.e., 180° 

  • In a rhombus, the two diagonals form four right-angled triangles that are congruent to each other

  • On joining the midpoint of the sides of a rhombus, you will get a rectangle

  • If you join the midpoints of half the diagonals, you will get another rhombus

  • There can be no circumscribing circle around a rhombus 

  • There can be no inscribing circle within a rhombus

  • If the shorter diagonal of a rhombus is equal to one of its sides, you will get two congruent equilateral triangles 

  • When a rhombus is revolved about the line that joins the midpoints of the opposite sides as the axes of rotation, a cylindrical surface with concave cones on both the ends is formed. 

  • When a rhombus is revolved around any of its sides as the axes of rotation, a cylindrical surface with a concave cone at one end and convex cone at another end are formed.

  • If the rhombus is revolved about its longer diagonal as the axis of rotation, then a solid having two cones attached to its bases is formed. In this case, the maximum diameter of the shape (solid) will be equal to the rhombus's shorter diagonal. 

  • When the rhombus is revolved about its shorter diagonal as the axis of rotation, then you will obtain a solid shape with two cones attached to its bases. The maximum diameter of the solid obtained in this case will be equal to the longer diagonal of the rhombus. 

  • When the midpoints of all the four sides of a rhombus are joined with each other, you will obtain a rectangle whose length and width will measure half of the value of prime diagonal. Moreover, the area of the rectangle formed in this case will be half of the rhombus.