Properties of Rhombus: Definition, Properties, and Formulae

Properties of Rhombus: A rhombus is a particular parallelogram because it meets the definition of a parallelogram, which is a quadrilateral having two sets of parallel sides. Furthermore, a rhombus, like a square, has all four sides that are equal. Thus rhombus is also known as a slanted square. A rhombus is considered a special parallelogram since the characteristics of a rhombus possess all the features of a parallelogram.

The two diagonals of a rhombus serve as its two lines of symmetry. An axis of symmetry is a line that splits an item into two equal halves. It produces a mirror-like reflection of the object’s two sides. A rhombus has reflection symmetry along each of its diagonals. Keep reading to learn about the rhombus angle properties, rhombus shape properties, and more.

Rhombus Definition: Characteristics of Rhombus

Before we discuss the rhombus, let us understand the characteristics of the rhombus, and what is a quadrilateral. In geometry, a quadrilateral is defined as a closed, two-dimensional shape that has four straight sides.

The sum of the interior angles of a quadrilateral is equal to 360°. There are mainly 6 types of quadrilateral:

1. Parallelogram
2. Trapezium
3. Square
4. Rectangle
5. Kite
6. Rhombus

A rhombus is a special type of parallelogram whose all four sides are equal. Thus, it is also called an equilateral parallelogram. The best example of a rhombus shape is a kite, with all its four sides equal. 

In the rhombus ABCD above, AB, BC, CD, and AD are the sides of the rhombus, and AC & BD are the diagonals. The length of AC and BD is d1 and d2, respectively. The two diagonals of the rhombus intersect at right angles as you can see in the figure.

Some characteristics of a rhombus are:

1. In a rhombus, opposite sides are parallel and the opposite angles are equal.
2. All the sides of a rhombus are equal in length.
3. Diagonals of a rhombus bisect each other at right angles.
4. The plural form of a rhombus is rhombi or rhombuses.

Other important Maths articles:

Properties of Rhombus Class 9

Understanding the properties of a rhombus for Class 9 student is one of the most important topics for CBSE Class 9 students as they are asked frequently in the final examination. Also, we have included all the important rhombus properties for Class 8 so that all Class 8 students also can benefit from them. You can read the rhombus properties here including rhombus angle properties and rhombus shape properties. Students also look for rhombus properties diagonals. Download them as a PDF provided below for offline access.

We have listed down all the important properties of the rhombus in points below:

1. All sides of a rhombus are equal.
2. The opposite sides of a rhombus are parallel.
3. Opposite angles of a rhombus are equal.
4. In a rhombus, diagonals bisect each other at right angles.
5. Diagonals bisect the angles of a rhombus. This is one of the most important properties of diagonals of rhombus.
6. The sum of two adjacent angles is equal to 180°.
7. The two diagonals of a rhombus form 4 right-angled triangles which are congruent to each other.
8. Rhombus lines of symmetry: There are only two lines of symmetry in a rhombus.
9. Rotational symmetry of rhombus: A rhombus has rotational symmetry of 180° (Order 2).
10. When you join the midpoint of all the 4 sides of a rhombus, it will form a rectangle. The length and width of the rectangle will be half the value of the main diagonal so that the area of the rectangle will be half of the rhombus.
11. You will get another rhombus when you join the midpoints of half the diagonal.
12. There is no circumscribing circle for a rhombus.
13. Also, there can be no inscribing circle within a rhombus.
14. When the shorter diagonal is equal to one of the sides of a rhombus, two congruent equilateral triangles are formed.
15. You will get a cylindrical surface having a convex cone at one end and concave cone at another end when the rhombus is revolved about any side as the axis of rotation.
16. You will get a cylindrical surface having concave cones on both the ends when the rhombus is revolved about the line joining the midpoints of the opposite sides as the axis of rotation.
17. You will get solid with two cones attached to their bases when the rhombus is revolving about the longer diagonal as the axis of rotation. In this case, the maximum diameter of the solid is equal to the shorter diagonal of the rhombus.
18. You will get solid with two cones attached to their bases when the rhombus is revolving about the shorter diagonal as the axis of rotation. In this case, the maximum diameter of the solid is equal to the longer diagonal of the rhombus.

Get Algebra Formulas from below:

Rhombus Formulas

We have presented you with the list of all the formulas for rhombus. The formulas are available for Area, Perimeter, Diagonal, and Side. Take the rhombus ABCD:

Sides: AB, BC, CD, and AD
Length of Each Side: a
Diagonals: AC, BD
Length of Diagonals: d1, d2

Students should note that problems related perimeter of rhombus formula are the most commonly asked questions. We have the following formulas:

Solved Problems on Rhombus

Here we have provided some of the questions with solutions related to the properties of rhombus:

Q1: What is the perimeter of a rhombus whose sides are all equal to 8 cm?

Solution: Side of rhombus = 8 cm (Given)
Since all the sides of a rhombus are equal, therefore,
Perimeter = 4 x side
P = 4 x 8 cm
= 32 cm
Hence, the perimeter of the rhombus is 32 cm.

Q2: Find the diagonal of a rhombus if its area is 121 cm² and the length of the longer diagonal is 22 cm.

Solution: Area of rhombus = 121 cm² (Given)
d₁ = 22 cm.
Area of the rhombus, A = (d₁ x d₂)/2, we get
121 = (22 x d₂)/2
121 = 11 x d₂
or 11 = d₂
So, the length of the other diagonal is 11 cm.

Q3: Find the perimeter of the following rhombus:

Solution: All the sides of a rhombus are congruent, so HO = (x + 2). And because the diagonals of a rhombus are perpendicular, triangle HBO is a right triangle. With the help of Pythagorean Theorem, we get,
(HB)² + (BO)² = (HO)²
x² + (x+1)² = (x+2)²
x² + x² + 2x + 1 = x² + 4x + 4
x² – 2x -3 = 0
Solving for x using the quadratic formula, we get:
x = 3 or x = –1. We can reject x = –1 since side of a rhombus cannot be negative.
∴ Side of the rhombus = x + 2
= 5
Hence, the perimeter of the rhombus HRMO is 5 x 4 units = 20 units.

Q4: The two diagonal lengths d₁ and d₂ of a rhombus are 5cm and 14 cm, respectively. Find its area.

Solution: Given:
Diagonal d₁ = 5cm
Diagonal d₂= 14 cm
Area of the rhombus, A = (d₁ x d₂)/2 square units
A = (5 x 14)/2
A = 70/2
A = 35 cm²
Therefore, the area of the rhombus = 35 square units.

Practice Questions on Properties of Rhombus

Here we have provided some practice questions related to the properties of rhombus for you to practice.

FAQs on Properties of Rhombus

Here we have provided some of the frequently asked questions:

Q1: What are the basic properties of a rhombus?

Ans: The basic properties of a rhombus are as follows:

 (i) The opposite angles are congruent.

 (ii) The diagonals intersect each other at 90 degrees.

 (iii) The diagonals bisect the opposite interior angles.

 (iv) The adjacent angles are supplementary. 

Q2: What is the formula for the area and perimeter of the rhombus?

Ans: Area of a rhombus = (d1.d2)/2, where d1 and d2 are the lengths of diagonals of the rhombus-

rhombus perimeter = 4 x Side of a rhombus 

Q3: Is Square a Rhombus?

Ans: Yes, geometrically, a square is also a rhombus whose all internal angles are 90 degrees. However, the vice-versa is not true. A rhombus may or may not be a square.

Q4: What are the angular properties of the rhombus?

Ans: The opposite angles of a rhombus are equal to each other. Also, the diagonals of a rhombus bisect the internal angles.

Q5: What are the symmetry properties of the rhombus?

Ans: A rhombus has 2 lines of symmetry which cuts it into two identical parts. The rhombus lines of symmetry are both its diagonals. Also, a rhombus has rotational symmetry. A rhombus has rotational symmetry of 180° (Order 2).

Q6: How many lines of symmetry does a rhombus have?

Ans: A rhombus has 2 lines of symmetry. The diagonals of a rhombus are its symmetry lines.

Q7: Are the diagonals of the rhombus equal?

Ans. No. In a rhombus, the diagonals bisect each other at perpendicular angles. They are not equal to each other in length. It means that the diagonals cut each other in half.

Practice Questions and Mock Tests for Maths (Class 8 to 12)

In this article, we have provided you with comprehensive information and all important points to learn about rhombus, like its characteristics, properties, formulae, solved examples and much more. It is time to start preparing for your Class 10 board exam. Refer to the links given in this article to learn the basic concepts on different Maths topics. Take our mock tests and practice more questions to master the Maths subject. 

We hope this detailed article helps you. If you have any queries regarding this article, reach out to us through the comment section below and we will get back to you as soon as possible.

Students can make use of NCERT Solutions for Maths provided by Embibe for their exam preparation.