Properties Of Rhombus?: A rhombus is a quadrilateral whose all four sides have the same length. It is a special kind of parallelogram whose diagonals intersect each other at 90°. This is one of the special properties of rhombus which is very helpful in many mathematical calculations.
A rhombus also called a diamond because of its diamond-like shape. Few examples of rhombus in our day-to-day life include kite, windows of a car, rhombus-shaped earrings, the structure of a building, mirrors, diamond cards in the deck of cards, etc. In this article, we have provided all the important properties of rhombus along with formulas related to rhombus. Read on to find out!
Rhombus Definition: Characteristics Of Rhombus
Before we discuss rhombus, let us understand 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:
- Parallelogram
- Trapezium
- Square
- Rectangle
- Kite
- Rhombus
A rhombus is a special type of parallelogram whose all four sides are equal. Thus, it is also called an equilateral parallelogram.
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:
- In a rhombus, opposite sides are parallel and the opposite angles are equal.
- All the sides of a rhombus are equal in length.
- Diagonals of a rhombus bisect each other at right angles.
- The plural form of a rhombus is rhombi or rhombuses.
Other important Maths articles:
Properties Of Rhombus
The properties of rhombus for Class 9 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 properties of rhombus for Class 8 so that all Class 8 students can benefit from them. You can read the properties here or download them as a PDF provided below for offline access.
We have listed down all the important properties of rhombus in points below:
- All sides of a rhombus are equal.
- The opposite sides of a rhombus are parallel.
- Opposite angles of a rhombus are equal.
- In a rhombus, diagonals bisect each other at right angles.
- Diagonals bisect the angles of a rhombus. This is one of the most important properties of diagonals of rhombus.
- The sum of two adjacent angles is equal to 180°.
- The two diagonals of a rhombus form 4 right-angled triangles which are congruent to each other.
- Rhombus lines of symmetry: There are only two lines of symmetry in a rhombus.
- Rotational symmetry of rhombus: A rhombus has rotational symmetry of 180° (Order 2).
- 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.
- You will get another rhombus when you join the midpoints of half the diagonal.
- There is no circumscribing circle for a rhombus.
- Also, there can be no inscribing circle within a rhombus.
- When the shorter diagonal is equal to one of the sides of a rhombus, two congruent equilateral triangles are formed.
- 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.
- 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.
- 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.
- 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.
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Rhombus Formulas
We have presented you 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
We have the following formulas:
| Area of Rhombus | Area, A = (d1.d2)/2 |
| Perimeter of Rhombus | Perimeter, P = 4a |
| Side of Rhombus | Side, a = P/4 |
| Diagonal of Rhombus | Diagonal, d1 = 2(A/d2) Diagonal, d2 = 2(A/d1) |
Solved Problems On Rhombus
Here we have provided some of the questions with solutions related to properties of a 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, 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, 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 rhombus = 35 square units.
Practice Questions On Properties Of Rhombus
Here we have provided some practice questions related to rhombus for you to practice.
| Q1: If the area of a rhombus is 48 cm2 and one of its diagonal is 5 cm. Find its altitude. Q2: ABCD is a rhombus in which the altitude from D to side AB bisects AB. Find the value of angle A and angle B. Q3: Show that area of a rhombus is half the product of its diagonals. Q4: Diagonal AC of a parallelogram ABCD bisects angle A. (a) Does angle A bisects angle C also? Give reasons. (b) Is ABCD a rhombus? Give reasons. Q5: If the length of each side of a rhombus is 8cm and one of its angles is 60° , then find the length of diagonals of rhombus. Q6: A rhombous sheet whose perimeter is 32m and whose one diagonal is10m long is painted on both side at the rate of ₹5 per meter square. Find the total cost of painting? Q7: The length of diagnols of a rhombus are in the ratio 5:4. The area of the rhombus is 2250 square cm. Find side of the rhombus. Q8: ABCD is a rhombus with intersecting point of diagonal is O and angle DAO = 45°. Find angle DCO. Q9: Prove that the diagonal of a rhombus bisect each other at right angles. Q10: If ABCD is rhombus and from D, an altitude is drawn to AB and it bisect the AB. Find the angles of the rhombus. |
FAQs On Properties Of Rhombus
Here we have provided some of the frequently asked questions:
A: 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.
A: Area of a rhombus = (d1.d2)/2, where d1 and d2 are the lengths of diagonals of the rhombus
Perimeter of rhombus = 4 x Side of rhombus
A: 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.
A: The opposite angles of a rhombus are equal to each other. Also, the diagonals of a rhombus bisect the internal angles.
A: 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 a rotational symmetry of 180° (Order 2).
A: A rhombus has 2 lines of symmetry. The diagonals of a rhombus are its symmetry lines.
Now you are provided with all the necessary information regarding rhombus and its properties. Practice more questions and master this concept.
Students can make use of NCERT Solutions for Maths provided by Embibe for your exam preparation.
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