## 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 =

^{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

**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.

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.

**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

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.

### FAQs (Frequently Asked Questions)

**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.

**What is the Perimeter of a Rhombus?**

The perimeter of a rhombus is the sum of all its sides. Since all the four sides of a rhombus are equal, the formula used to find the perimeter of a rhombus is: Perimeter = a + a + a + a = 4a, where ‘a’ represents the side length of the rhombus.

**What is the Area and Perimeter of a Rhombus?**

The area of a rhombus is the space occupied by it. This is calculated with the help of different formulas which depend on the type of dimensions given and is expressed in square units.

- When the base and height of a rhombus are known, then the area of rhombus = base × height.
- When the diagonals of a rhombus are known, then the area = (diagonal 1 × diagonal 2)/2.

The perimeter of a rhombus is the total length of its boundary. This is calculated with the help of different formulas which depend on the given dimensions and is expressed in linear units.

- When one side of a rhombus is known, the perimeter = 4 × side
- When the two diagonals of a rhombus are known, the

**What is the Perimeter of a Rhombus Formula When Side Length is Given?**

Since all the 4 sides of a rhombus are equal, its perimeter is obtained by multiplying its side by 4. The formula to calculate the perimeter of a rhombus with side ‘a’ is: P = 4a.

**How to Find the Side Length When the Perimeter of a Rhombus is Given?**

We know that the perimeter of a rhombus with side length ‘a’ is calculated with the help of the formula, P = 4a. Thus, the side length of the rhombus can be obtained by dividing its perimeter by 4.

## Perimeter of Rhombus

The perimeter of a rhombus is the sum of all its sides. Rhombus is a quadrilateral in which all four sides are of the same measure. A rhombus is always a parallelogram but a parallelogram may not necessarily be a rhombus always. Let us study more about the perimeter of rhombus in this article.

### What is the Perimeter of a Rhombus?

The perimeter of a rhombus is the total measure of its boundary and it is calculated by adding the length of all its sides. Since all the four sides of a rhombus are equal, the basic formula to find the perimeter of a rhombus is: Perimeter = 4a; where ‘a’ is the side of the rhombus. The perimeter of a rhombus is expressed in linear units like inches, yards, centimeters, and so on.

### Properties of a Rhombus

There are some basic properties that help us to identify a rhombus. Observe the following rhombus ABCD to relate to its properties given below.

We can identify and distinguish a rhombus with the help of the following properties:

- All four sides are equal.
- The opposite sides are parallel.
- The opposite angles are equal.
- The sum of any two adjacent angles is 180
^{o}. - The diagonals bisect each other at right angles.
- Each diagonal bisects the vertex angles.

Now, let us read about the formulas that are used to find the perimeter of a rhombus in different cases. The formulas differ according to the known dimensions.

### Perimeter of Rhombus Formula With Sides

As discussed earlier, the perimeter of a rhombus is the sum of the lengths of all its sides. We know that the sides of a rhombus are of equal lengths. Let us consider a rhombus of side length ‘a’. Then, the perimeter of the rhombus is a + a + a + a which is 4a. Thus, the perimeter of the rhombus formula is: P = 4a.

**Example: **Find the perimeter of a rhombus that has a side length of 10 units.

**Solution:**

The side length of the given rhombus, a = 10 units.

Its perimeter = 4a = 4 × 10 = 40 units.

### Perimeter of Rhombus Formula With Diagonals

When the length of the diagonals of a rhombus is known, we find the side length of the rhombus using the Pythagoras theorem. Here, we make use of the following properties of a rhombus:

- A rhombus is divided into 4 congruent right-angled triangles by its two diagonals.
- The diagonals bisect each other at right angles.

Let us consider a rhombus ABCD with diagonals p and q and with side length ‘a’.

Let us take triangle AOD. Since the diagonals bisect at right angles, AO can be written as p/2 and OD can be written as q/2. Now, if we apply the Pythagoras theorem for triangle AOD, we get

**Note:** We do not need to remember this formula. We can use the Pythagoras theorem to find the side length of the rhombus using the diagonals and then we can apply the perimeter of rhombus formula to be, P = 4 × side length.

### Perimeter of Rhombus Examples

**Example 1:** Find the perimeter of a rhombus with diagonals 8 inches and 6 inches respectively.

**Solution:**

**Method 1:**

The lengths of the diagonals of the given rhombus are, p = 8 inches and q = 6 inches.

Using the perimeter of rhombus formula using diagonals,

By Pythagoras theorem,

a^{2} = 3^{2} + 4^{2} = 25

a = 5 inches.

Thus, the perimeter of the rhombus is, 4a = 4 × 5 = 20 inches.

**Answer:** The perimeter of the given rhombus = 20 inches.

### Real-life Applications of the Perimeter of a Rhombus

The word perimeter is a combination of two Greek words: “Peri,” which means surrounding or boundaries of a surface or an object, and “Meter,” which means measurement of the surface or object, so perimeter means **the total measurement of boundaries of a given surface**.

With this information, we can use the perimeter of a rhombus in numerous real-life applications. Various examples *are given below:*

- For example, we can use the perimeter of a rhombus to calculate the distance of a pitcher’s spot from the striker in baseball if the whole pitch is shaped like a rhombus.
- The perimeter formula is also helpful in designing tables and cupboards having a rhombus shape.
- It is also helpful in the construction of rhombus-shaped offices and rooms.

Example 1:

If the length of one side of a rhombus is 11 cm, what will be the length of the rest of the sides?

Solution:

We know that **all the sides of a rhombus are equal in length**, so the length of the rest of the three sides is also 11 cm each.

Example 2:

Calculate the perimeter of a rhombus for the figure given below.

Solution:

We are given the length of one side of a rhombus, and we know that **all the sides are equal in length**.

Perimeter of the rhombus Perimeter of the rhombus

Example 3:

If the perimeter of a rhombus is 80cm, what will be the length of all the sides of the rhombus?

Solution:

We are given the perimeter of the rhombus. We can calculate the length of each side of a rhombus by** using the perimeter formula**:

Example 4:

If the length of the diagonals of a rhombus is 9 cm and 11cm, what will be the perimeter of the rhombus?

Solution:

We are given the length of the two diagonals of the rhombus: let “a” and “b” be the two diagonals of the rhombus. Then, we can calculate the perimeter of the rhombus by **using the formula given below**.

### Example 5:

A rhombus has an area of 64cm^{2} , and the length of one diagonal of the rhombus is 8cm . What will be the perimeter of the rhombus?

### Solution:

Let diagonal “a” = 8cm and we have to find “b”

**Answer Key**

1. We know that **all sides of a rhombus are equal in length**. If the length of one side of the rhombus is 20 cm, then the length of the remaining three sides will also be the same, i.e., 20 cm.

3. We are given the lengths of the two diagonals of the rhombus. Let “a” and “b” be the two diagonals. Then, we can calculate the perimeter and area of the rhombus by **using the values of the diagonals**.

Perimeter of a rhombus =30cm approx.

**Question 1: Find the perimeter of a rhombus whose side is 8 cm.**

**Solution:**

Given that side s = 8 cm

Perimeter of Rhombus is given by : 4*s

So, Perimeter (P) = 4 * 8 cm = 32 cm

**Question 2: Find the side length of a rhombus whose perimeter is given as 36cm.**

**Solution:**

Given Perimeter(P) = 36 cm

P = 4 * s

=> s = P/4

So, s = 36/4 = 9cm

**Question 3: Find the perimeter of the rhombus given the diagonal lengths are 6 cm and 8 cm respectively.**

**Solution:**

When diagonal lengths are given :

Given a = 6 cm, b = 8cm

Perimeter(P) = 2* √(a

^{2}+ b^{2}) = 2* √(36 + 64) = 2 * 10 = 20 cm

**Question 4: Find the length of horizontal diagonal given the side length as 13cm and vertical diagonal length as 24 cm.**

**Solution:**

Since the diagonal bisect at right angles:

Given b = 24 cm and s = 13 cm, a = ?

side(s) is given as

s = (√(a

^{2}+ b^{2}))/22 * s = (√(a

^{2}+ b^{2}))26 = (√(a

^{2}+ 576))On squaring both sides, 676 = a

^{2}+ 576=> a

^{2}= 100=> a= 10cm

**Question 5: Find the area of the rhombus whose diagonal are of lengths 24cm and 10 cm.**

**Solution:**

Given a = 24 and b = 10cm

Area of the Rhombus is given by A = 1/2 * a * b

= 1/2 * 24 * 10

= 60 cm

^{2}

**Question 6: Find the perimeter of a rhombus whose side is 2.5 cm.**

**Solution:**

Given that side s = 8 cm

Perimeter of Rhombus is given by: 4*s

So, Perimeter (P) = 4 * (2.5) cm = 10 cm

**Question 7: Find the side length of a rhombus whose perimeter is given as 48cm.**

**Solution:**

Given Perimeter(P) = 48 cm

P = 4 * s

=> s = P/4

So, s = 48/4 = 12cm

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