## Formulas of Rhombus

Wondering how to find the perimeter of a rhombus? Just like any polygon, the perimeter of the rhombus is the total distance around the outside, which can be simply calculated by adding up the length of each side. In the case of a rhombus, all four sides are of similar length, thus the perimeter is four times the length of aside. Or as a formula: Perimeter of rhombus = 4 a = 4 × side. Here, ‘a’ represents each side of a rhombus.

### About the Perimeter of Rhombus Formula

A rhombus is a 2-dimensional (2D) geometrical figure which consists of four equal sides. Rhombus has all sides equal and its opposite angles are equivalent in measurement. Let us now talk about the rhombus formula i.e. area and perimeter of the rhombus.

### The Perimeter of a Rhombus

The perimeter is the sum of the length of all 4 sides. In the rhombus all sides are equal.

Thus, the Perimeter of rhombus = 4 × side

So, P = 4s

In which,

S = length of a side of a rhombus

### Area of Rhombus Formula

The area of a rhombus is the number of square units in the interior of the polygon. The area of a rhombus can be identified in 2 ways:

i) Multiplying the base and height as rhombus is a unique kind of parallelogram.

Area of rhombus = b × h

In which,

B = base of the rhombus

H = height of the rhombus

ii) By determining the product of the diagonal and dividing the product by 2.

Area of rhombus formula = \[\frac {1}{2}\] × d1 × d2

In which,

d_{1} × d_{2} = diagonal of the rhombus

### Derivation of Area of Rhombus

Let MNOP is a rhombus whose base MN = b, PN ⊥ MO, PN is a diagonal of rhombus = d_{1}, MO is diagonal of rhombus = d_{2}, and the altitude from O on MN is OZ, i.e., h.

Area of Rhombus MNOP = 2 Area of ∆ MNO

= 2 × \[\frac {1}{2}\] MN × OP sq units.

= 2 × \[\frac {1}{2}\] b × h sq. units

= base × height sq. units

Area of rhombus = 4 × Area of ∆ MZP

= 4 × \[\frac {1}{2}\] × MZ × ZN sq. units

= 4 × \[\frac {1}{2}\] × d_{2} × \[\frac {1}{2}\] d_{1} sq. units

Thus,

= 4 × \[\frac {1}{8}\] d_{1} × d_{2} sq. units

= \[\frac {1}{2}\] × d_{1} × d_{2}

Hence, the area of a rhombus

= \[\frac {1}{2}\] (product of diagonals) sq. units.

### Fun Facts

A rhombus consists of an inscribed circle

In a rhombus, all sides are equal, just as a rectangle has all angles equal.

A rhombus has opposite angles equivalent to each other, while a rectangle has opposite sides equal.

### Finding the Perimeter of a Rhombus when only Diagonals are known

In many questions, you will see that the length of the sides of the rhombus is not given. Instead, the question will provide you with the length of its diagonals. You can use the diagonals to find the side of the rhombus and calculate its perimeter. Here is how you can do it:

Consider a rhombus PQRS with diagonals a and b, and center O.

Pick a triangle, say POS. Since the diagonals of a rhombus bisect each other at right angles, the length of OP will be \[\frac {a}{2}\] and OS will be \[\frac {b}{2}\].

Assume that the side of the rhombus is X.

Using Pythagoras theorem in triangle POS, we get:

X_{2} = \[\frac {a} {2}^2 + \frac {b} {2}^2 \], which gives us,

X_{2} = \[\frac {(a^2+b^2)} {4} \]

X = \[\sqrt \frac {(a^2+b^2)} {2} \]

Substitute the value of X in the perimeter of the rhombus formula i.e. 4X. You will get:

4 × \[\sqrt \frac {(a^2+b^2)} {2} \]

6. So, the perimeter of the rhombus will be equal to 2 × \[\sqrt {(a^2+b^2)} \].

### Solved Examples

1. Evaluate the area of the rhombus MNOP having each side equal to 15 cm and one of its diagonals equal to 18 cm.

Ans:

Given:- MNOP is a rhombus in which MN = NO = OP = PM = 15 cm

MO = 18 cm

Thus, MZ = 9 cm

In ∆ MZP,

MP^{2} = MZ^{2} + ZP^{2}

⇒ 152 = 92 + ZP^{2}

⇒ 225 = 81 + ZP^{2}

⇒ 144 = ZP^{2}

⇒ ZP = 12

Hence, NP = 2 P

= 2 × 12

= 24 cm

Now, to find out the area of the rhombus, we will apply the formula i.e.

= \[\frac {1}{2}\] × d_{1} × d_{2}

= \[\frac {1}{2}\] × 18 × 24

= 216 cm^{2
}

2. Find the perimeter of a rhombus MNOP whose diagonals measure 20 cm and 24 cm respectively?

Ans:

Given: d_{1} = 20 cm

d_{2} = 24 cm

MZ= \[\frac {20}{2}\] = 10cm

NZ= \[\frac {24}{20}\]= 12 cm

∠MZP = 90°

Now applying the Pythagorean Theorem, we know that

MN^{2} = MZ^{2} + NZ^{2}

MN = \[\sqrt {(100+144)} \]

= 15.62 cm

Since, MN = NO = OP = MP,

Therefore, Perimeter of MNOP = 15.62 × 4 = 62.48 cm.

### Properties of a Rhombus

Identifying a rhombus is not that difficult. There are some properties of a rhombus that will help you determine whether a given figure is a rhombus or not. If a shape meets the following conditions, then it is a rhombus:

All the sides of a rhombus are equal.

The opposite sides of a rhombus are parallel to each other.

All the opposite angles in a rhombus will be equal.

The diagonals of a rhombus bisect each other at 90 degrees i.e. right angles.

When you add any two adjacent angles of a rhombus, the sum should be equal to 180 degrees.

### Revising the Perimeter of Rhombus Formula - Explanation, Area, Solved Examples and FAQs

The Perimeter of Rhombus Formula is one of the most important formulas of Mathematics. It comes under mensuration, which is a crucial chapter of Maths. You must have a clear understanding of the Perimeter of Rhombus Formula - Explanation, Area, Solved Examples and FAQs to ensure a good score in your finals. Once you have a firm grasp of the perimeter and area of a rhombus, you will be able to solve any question related to this concept. That is why you should practice and revise the Perimeter of Rhombus Formula - Explanation, Area, Solved Examples and FAQs thoroughly. Here are some revision tips:

First, study the Perimeter of Rhombus Formula - Explanation, Area, Solved Examples and FAQs provided to understand the topic clearly.

Go through the textbook explanations of the perimeter and area of a rhombus thoroughly.

Practice as many questions related to the perimeter and area of a rhombus as you can to become more proficient at solving mensuration problems.

Use different reference books and pick out the important questions based on the perimeter and area of a rhombus to understand the different types of questions that can come in your final Maths exam.

Visit Vedantu’s e-learning platform through our website or mobile application to study the Perimeter of Rhombus Formula - Explanation, Area, Solved Examples and FAQs. Every explanation and question provided by Vedantu is curated by some of the best subject matter experts to ensure accuracy and high quality.

Go through solved examples from the textbook or reference books to learn how to solve different kinds of questions based on the area and perimeter of a rhombus.

Once you have solved all the textbook questions, find out questions from previous year question papers and sample papers to understand the pattern and difficulty level of the exam.

**1. What is a Rhombus?**

A rhombus is a special kind of parallelogram with its entire sides equal. A square is a kind of rhombus with all of the angles being equal and also all of the sides. Moreover, both rhombuses and squares have perpendicular diagonal bisectors that divide each diagonal into two equal segments, and also divide the quadrilateral into four equal right triangles.

Having said that, we know the Pythagorean Theorem would work sufficiently in this situation, using half of each diagonal as the two legs of the right triangle.

**2. What are the properties of a Rhombus?**

**Following are the properties of a rhombus:-**

Rhombus has all sides equal.

The opposite angles of a rhombus are equal.

The Sum of adjacent angles are supplementary i.e. (∠N + ∠O = 180°).

If one angle of a rhombus is right, then all angles are right.

Each diagonal splits it into two congruent triangles.

In a rhombus, diagonals intersect each other and are perpendicular to each other.

**3. What are the dual properties of a Rhombus?**

**Note that the dual polygon of a rhombus is a rectangle:**

A rhombus consists of an axis of symmetry across each pair of opposite vertex angles, whereas a rectangle has an axis of symmetry across each pair of opposite sides.

The diagonals of a rhombus bisect at equal angles. On the other hand, the diagonals of a rectangle are equivalent in length.

The geometrical shape formed by connecting the midpoints of the sides of a rhombus is a rectangle, and reciprocally.

**4. From where can I learn the Perimeter of Rhombus Formula - Explanation, Area, Solved Examples, and FAQs?**

You can learn the Perimeter of Rhombus Formula - Explanation, Area, Solved Examples, and FAQs from Vedantu’s online learning platform. To start learning, visit Vedantu.com or download our mobile application from the play store for android and the app store for iPhone. You don’t have to pay any registration fee to learn the Perimeter of Rhombus Formula - Explanation, Area, Solved Examples, and FAQs on Vedantu. Moreover, we provide you with plenty of study materials to learn various formulas of Maths. You can use these resources to learn formulas of Relations and Functions, Trigonometry, Matrices, Determinants, and much more for absolutely free only on Vedantu’s educational platform.

**5. How will the perimeter of Rhombus Formula - Explanation, Area, Solved Examples and FAQs help me?**

Learning the Perimeter of Rhombus Formula - Explanation, Area, Solved Examples and FAQs will help you solve many mensuration problems. You can use these formulas to determine the perimeter and area of a rhombus. By solving questions with these formulas, you will get a better understanding of a rhombus. Your exam question paper might contain questions based on the perimeter or area of a rhombus. So, the Perimeter of Rhombus Formula - Explanation, Area, Solved Examples and FAQs will aid you in scoring well in your final exams.