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## Area Of Rhombus

In geometry, a rhombus is a special type of parallelogram in which two pairs of opposite sides are congruent. That means all the sides of a rhombus are equal. Students often get confused with square and rhombus. The main difference between a square and a rhombus is that all the internal angles of a square are right angles, whereas they are not right angles for a rhombus. In this article, you will learn how to find the **area of a rhombus** using various parameters such as diagonals, side & height, and side and internal angle, along with solved examples in each case.

## What is the Area of a Rhombus?

The area of a rhombus can be defined as the amount of space enclosed by a rhombus in a two-dimensional space. To recall, a rhombus is a type of quadrilateral projected on a two dimensional (2D) plane, having four sides that are equal in length and are congruent.

## Area of Rhombus Formula

Different formulas to find the area of a rhombus are tabulated below:

Formulas to Calculate Area of Rhombus | |
---|---|

Using Diagonals | A = ½ × d_{1} × d_{2} |

Using Base and Height | A = b × h |

Using Trigonometry | A = b^{2} × Sin(a) |

Where,

- d
_{1 }= length of diagonal 1 - d
_{2 }= length of diagonal 2 - b = length of any side
- h = height of rhombus
- a = measure of any interior angle

## Derivation for Rhombus Area Formula

Consider the following rhombus: ABCD

Let O be the point of intersection of two diagonals AC and BD.

The area of the rhombus will be:

A = 4 × area of ∆ AOB

= 4 × (½) × AO × OB sq. units

= 4 × (½) × (½) d_{1} × (½) d_{2} sq. units

= 4 × (1/8) d_{1} × d_{2} square units

= ½ × d_{1} × d_{2}

Therefore, the **Area of a Rhombus = A = ½ × d _{1} × d_{2}**

Where d_{1} and d_{2} are the diagonals of the rhombus.

## How to Calculate Area of Rhombus?

The methods to calculate the area of a rhombus are explained below with examples. There exist three methods for calculating the area of a rhombus, they are:

- Method 1: Using Diagonals
- Method 2: Using Base and Height
- Method 3: Using Trigonometry (i.e., using side and angle)

### Area of Rhombus Using Diagonals: Method 1

Consider a rhombus ABCD, having two diagonals, i.e. AC & BD.

**Step 1:** Find the length of diagonal 1, i.e. **d _{1}**. It is the distance between A and C. The diagonals of a rhombus are perpendicular to each other by making 4 right triangles when they intersect each other at the centre of the rhombus.

**Step 2:** Find the length of diagonal 2, i.e. **d _{2}** which is the distance between B and D.

**Step 3:** Multiply both the diagonals, d_{1}, and d_{2}.

**Step 4:** Divide the result by 2.

The resultant will give the area of a rhombus ABCD.

Let us understand more through an example.

**Example 1**: **Calculate the area of a rhombus having diagonals equal to 6 cm and 8 cm.**

**Solution:**

Given that,

Diagonal 1, d_{1} = 6 cm

Diagonal 2, d_{2} = 8 cm

Area of a rhombus, A = (d_{1} × d_{2}) / 2

= (6 × 8) / 2

= 48 / 2

= 24 cm^{2}

Hence, the area of the rhombus is 24 cm^{2}.

### Area of Rhombus Using Base and Height: Method 2

**Step 1: **Find the base and the height of the rhombus. The base of the rhombus is one of its sides, and the height is the altitude which is the perpendicular distance from the chosen base to the opposite side.

**Step 2:** Multiply the base and the calculated height.

Let us understand this through an example:

**Example 2:** **Calculate the area of a rhombus if its base is 10 cm and height is 7 cm.**

**Solution:**

Given,

Base, b = 10 cm

Height, h = 7 cm

Area, A = b × h

= 10 × 7 cm^{2}

A = 70 cm^{2}

### Area of Rhombus Using Trigonometry: Method 3

This method is used to calculate the area of the rhombus when the side and one of its internal angles are given.

**Step 1:**Square the length of any of the sides.**Step 2:**Multiply it by Sine of one of the angles.

Let us see how to find the area of a rhombus using the side and angle in the below example.

**Example 3:** **Calculate the area of a rhombus if the length of its side is 2 cm and one of its angles A is 30 degrees.**

**Solution:**

Given,

Side = s = 2 cm

Angle A = 30 degrees

Square of side = 2 × 2 = 4

Area, A = s^{2} × sin (30°)

A = 4 × 1/2

A = 2 cm^{2}

### Solved Problem on Area of Rhombus Formula

**Question: Find the area of the rhombus having each side equal to 17 cm and one of its diagonals equal to 16 cm.**

**Solution:**

ABCD is a rhombus in which AB = BC = CD = DA = 17 cm

Diagonal BD = 16 cm (with O being the diagonal intersection point)

Therefore, BO = OD = 8 cm

In ∆ AOD,

AD^{2} = AO^{2} + OD^{2}

⇒ 17^{2} = AO^{2} + 8^{2}

⇒ 289 = AO^{2} + 64

⇒ 225 = AO^{2}

⇒ AO = 15 cm

Therefore, AC = 2 × AO

= 2 × 15

= 30 cm

Now, the area of the rhombus

= ½ × d_{1} × d_{2}

= ½ × 16 × 30

### Practice Questions

- Find the height of the rhombus, whose area is 175 cm² and perimeter is 100 cm.
- Calculate the area of a rhombus with a side of 5 cm, and one of the internal angles is 120 degrees.
- If the area of a rhombus is 143 sq. units and one of its diagonal is 26 units, find the other diagonal.

## Frequently Asked Questions

### What is a Rhombus?

A rhombus is a type of quadrilateral whose opposite sides are parallel and equal. Also, the opposite angles of a rhombus are equal and the diagonals bisect each other at right angles.

### What is the Formula for Area of a Rhombus?

To calculate the area of a rhombus, the following formula is used:

A = ½ × d_{1} × d_{2}

### How to Find the Area of a Rhombus When the Side and Height are Given?

To find the area of a rhombus when the measures of its height and side are given, use the following formula:

A = Base × Height

### What is the Formula for Perimeter of a Rhombus?

The formula to calculate the perimeter of a rhombus of side “a” is:

P = 4a units

### How to find the area of a rhombus if one of its sides and an included angle are given?

If “a” be its sides and “θ” is an included angle, then the formula is:

Area of a Rhombus = a^{2} sin θ square units.

### Find the area of a rhombus if the diagonal measures 6cm and 4cm.

We know that, Area of Rhombus = (½) × Diagonal 1 × Diagonal 2

Substituting the values, we get

A = (½) × 4 × 6 = 12 cm^{2}.

### Is the area of a rhombus the same as the area of a square?

No, the area of a rhombus is not the same as the area of a square.

### What is the difference between the area of a rhombus and the area of a square?

The area of a square is the square of its side, whereas the area of a rhombus is the half the product of diagonal 1 and diagonal 2.

## What Is the Rhombus Formula?

The formulas for rhombus are defined for two features, an area of the rhombus and the perimeter of a rhombus. These rhombus formulas can be expressed as:

### Perimeter of a Rhombus Formula

The perimeter of a rhombus is the sum of its four sides, thus the perimeter of the rhombus formula = 4 × a, where ‘a’ is the side.

### Area of a Rhombus Formula

To find the area of a rhombus, we find the total area enclosed by a rhombus. The area of the rhombus can be calculated by the following rhombus formulas:

- When length of the diagonals are given, Area of a rhombus formula = 1/2 × d1

× d2

- When the length of the side of the rhombus and interior angle are given, Area of rhombus formula = a
^{2}× sin x

where,

- d1

and d2

- are the diagonals,
- a is the length of the side of the rhombus,
- x is the interior angle

Let us understand the rhombus formula better using a few solved examples.

## Examples on Rhombus Formulas

**Example 1: **Find the area of the rhombus of diagonal lengths 10 and 8 units using the rhombus formula.

**Solution:**

To find:

Area of the rhombus.

Using the rhombus formula,

Area of the rhombus = 1/2 × d1 × d2

Area of the rhombus = 1/2 × 10 × 8

Area of the rhombus = 40

**Answer:** **The area of the rhombus is 40 square units.**

**Example 2: **Using the rhombus formula find the area of the rhombus with an interior angle of 30 degrees and length of the side 5in.

**Solution:**

To find: Area of the rhombus.

Interior angle(a) = 30 degrees, side length = 5 in.

Using the rhombus formula,

Area of the rhombus = a^{2} × sin x

Area of the rhombus = 5^{2} × sin 30

Area of the rhombus = 25 × 1/2

Area of the rhombus = 12.5

**Answer: The area of the rhombus is 12.5 unit inches.**

**Example 3**:Arhombus-shaped tile is placed at the playground of length 16 units. Can you find the perimeter of the tile using the rhombus formula?

**Solution:**

Given:

Length of the tile = 16 units.

Since all sides of a rhombus are equal, all four sides are equal to 16 units.

Perimeter = 4 × side = 4 × 16 = 64 units

**Answer: **The perimeter of the tile is 64 units.

## FAQ’s on Rhombus Formulas

### What Are the Rhombus Formulas?

The formulae for rhombuses are defined for two parameters, area and perimeter:

### What Is the Rhombus Formula When the Two Diagonals are Given?

The area of a rhombus when diagonals are given is calculated by using the following rhombus formula.

## What is Area of a Rhombus?

The area of a rhombus can be defined as the amount of space enclosed by a rhombus in a two-dimensional space. It depicts the total number of unit squares that can fit into it and it is measured in square units (like cm^{2}, m^{2}, in^{2}, etc). Rhombus is a parallelogram with the opposite sides parallel, the opposite angles equal, and the adjacent angles supplementary. The following are the properties of the shape.

- A rhombus is an equilateral quadrilateral because all the sides have equal lengths.
- The diagonals of a rhombus bisect each other at right angles.
- The diagonals are angle bisectors.
- The area of a rhombus can be found in different ways: using base and height, using diagonals, and using trigonometry.

## Area of a Rhombus Formula

Different formulas can be used to calculate the area of a rhombus depending upon the parameters known to us. The different formulas followed for the calculation of the area of a rhombus are,

- Using base and height
- Using diagonals
- Using trigonometry

### Formula for Area of Rhombus When Base and Height Are Known

Rhombus is a parallelogram. We know that the area of a parallelogram is given by multiplying base and height sq units. The same is applied to the rhombus as well.

Area of a Rhombus = base × height sq units

**Example: Find the area of a rhombus having a side length of 7 inches and the height of the rhombus of 10 inches.**

**Solution:** As we know, Area = base × height units^{2}

⇒ Area = 7 × 10 inches^{2}

⇒ Area = 70 inches^{2}

### Formula for Area of Rhombus When Diagonals Are Known

The area of a rhombus is equal to half the product of the lengths of the diagonals. The formula to calculate the area of a rhombus using diagonals is given as,

Area = (d_{1} × d_{2})/2 sq. units, where, d_{1} and d_{2} are the diagonals of the rhombus.

Consider the Rhombus ABCD. Let E be the point of intersection of two diagonals. We make the following observations,

- Four sides are congruent.
- The diagonals bisect each other.
- Four interior angles with opposite angles equal. ⇒ ∠ A = ∠C and ∠B = ∠D
- The two diagonals are AC and BD.

The area of the rhombus ABCD = Area of ∆ ADC + Area of ∆ ABC

Area of the rhombus = 2 × Area of ∆ ABC —(1) (∵ ∆ ABC congruent to ∆ ADC)

Area of ∆ ABC

= 1/2 × Base × Height

= 1/2 × AC × BE

= 1/2 × AC × 1/2 × BD (∵BE = BD/2)

= 1/4 (AC × BD) — (2)

Area of the Rhombus ABCD

Area = 2 × 1/4 × AC × BD = 1/2 × AC × BD (From (1) and (2))

⇒ Area = 1/2 × diagonal 1 × diagonal 2

∴ Area of a Rhombus = 1/2 × diagonal 1 × diagonal 2 units^{2}

### Formula for Area of Rhombus When Side and Angles Are Known

We apply the concept of trigonometry while calculating the area when sides and angles are known. We can use any angle because either the angles are equal or they are supplementary, and supplementary angles have the same sine. Area of a rhombus using side and angle is given as,

Area of a Rhombus = side^{2 }× sin(A) sq. units, where ‘A’ is an interior angle.

**Example:** What is the area of a rhombus if the length of its side is 4 yards and one of its angle A is 30º.

Solution: As we know, area of rhombus = s^{2 }× sin(30º)

Area of rhombus = s^{2 }× sin(30º) = 4^{2} × 1/2

⇒ Area of rhombus = 16 × 1/2 = 8 sq yards

## How to Calculate the Area of a Rhombus?

The different methods to calculate the area of a rhombus are explained below. There are three methods for calculating the area of a rhombus, given as:

- Method 1: Using Base and Height
- Method 2: Using Diagonals
- Method 3: Using Trigonometry

**Area of Rhombus Using Base and Height**

**Step 1:**Find and note the base and the height of the given rhombus. The base is one of the sides of a rhombus, while the height is the perpendicular distance from the chosen base to the opposite side.**Step 2:**Multiply the base and height.

The resultant value will give the area of a rhombus.

**Area of Rhombus Using Diagonals**

Consider a rhombus ABCD, having two diagonals, i.e. AC & BD.

**Step 1:**Find the length of both the diagonals, diagonal 1 and diagonal 2.**Step 2:**Multiply both the lengths, d1, and d2.**Step 3:**Divide the result by 2.

The resultant value will give the area of a rhombus ABCD.

**Area of Rhombus Using Trigonometry**

**Step 1:**Square the length of any of the sides.**Step 2:**Multiply it by the sine of any one of the angles.

The resultant value will give the area of a rhombus.

**Example:** Consider the rhombus ABCD. AB, BC, CD, DA are the congruent (equal) sides. AC and BD are the diagonals, and they meet at E. Given CD = 17 feet and AE = 8 feet. Find the area of ABCD.

Now we know diagonal 1, AC = 16 feet.

Next, we need to calculate BD.

BD = BE + ED = 2 × BE

We still have an unknown, BE.

Pythagorean theorem states that

BC^{2} = BE^{2} + EC^{2}

BC = 17 feet (∵ CD =BC as all the sides are congruent)

EC = 8 feet (∵ AE = EC as the diagonals bisect each other)

17^{2} = BE^{2}+ 8^{2}

⇒ BE^{2} = 289 – 64

= 225

∴ BE = 15 feet and BD = 30 feet

Lets substitute all the values in the area of the rhombus formula.

Area of the Rhombus = 1/2 × d_{1} × d_{2} sq units

= 1/2 × BD × AC sq feet

= 1/2 × 30 × 16 sq feet

⇒ Area of rhombus = 240 sq feet

**Tips and Tricks:**

- Remember that the height is not the same as the length of the side of a rhombus.
- The area of a rhombus can be found in three ways: when diagonals are given, when an angle and a side are given, when angle and height are given.
- Use the Pythagorean theorem to find the second diagonal if the measures of one diagonal and side are given.

## Examples on Area of Rhombus

**Example 1:** Using the area of a rhombus formula, find the area of the rhombus given in the figure below.

**Solution:**

Area of a rhombus= 1/2 × BD × AC

BD = 2 × BE

= 2 × 8

= 16 yards

AC = 2 × AE = 2 × 10 = 2 yards

⇒ Area = 1/2 × 16 × 20

= 8 × 20

= 160 yards^{2}

**Answer:** Area of the rhombus = 160 yards^{2}

**Example 2:** The sides of a rhombus ABCD measure 5 inches, while the length of its one diagonal AC is 8 in. Calculate its area.

**Solution:**

Given a side and a diagonal, we will find the other diagonal. Let O be the point of intersection of diagonals.

Area = (AC × BD)/2 sq inches

⇒ AC = 8 in

⇒ AO = 4 in (∵ AO = 1/2 AC)

To find the other diagonal BD, consider AOD.

By Pythagorean theorem, AD^{2 }= AO^{2} + OD^{2}

⇒ 25^{2} = 4^{2} + OD^{2}

⇒ OD^{2} = 25 −16

⇒ OD^{2} = 9

⇒ OD = 3 in

⇒ BD = 6 in (∵BD = 2 × OD)

Area = (8 × 6) ÷ 2 sq inches

Area = 24 sq inches

**Answer:** Area of the rhombus = 24 sq inches

**Example 3:** The area of a rhombus is given as 256 square units. If the length of one of the diagonals is 8 units, find the length of its other diagonal.

**Solution:**

We know area of a rhombus can be calculated using diagonals as,

Area = (1/2) × (diagonal_{1} × diagonal_{2})

⇒ 256 = (1/2) × (8 × diagonal_{2})

diagonal_{2} = (256 × 2)/8

diagonal_{2} = 64

**Answer:** Length of the other diagonal of the given rhombus = 64 units

## FAQs on Area of Rhombus

### What is the Area of Rhombus?

The area of a rhombus is the total amount of space enclosed or encompassed by a rhombus in a two-dimensional plane. It is expressed in square units(like cm^{2}, m^{2}, in^{2}, etc).

### What is the Formula of Finding the Area of a Rhombus?

Different formulas can be used to calculate the area of a rhombus depending upon the parameters known to us. Using base and altitude the formula is given as, Area of a Rhombus = base × height sq units. Area of rhombus using diagonals is: Area = (d_{1} × d_{2})/2 sq. units, where d_{1} and d_{2} are the diagonals of the rhombus. Applying the concept of trigonometry using side and angle, we can follow the formula: Area of a Rhombus = side^{2 }× sin(A) sq. units, where ‘a’ is an interior angle.

### How do you Find the Side of a Rhombus With the Diagonals?

The area of a rhombus can be calculated using the lengths of diagonals. The formula to find area in this case is given as: Area = (d_{1} × d_{2})/2 sq. units, where d_{1} and d_{2} are the diagonals of the rhombus.

### Are the Areas of Rhombus and Square Equal?

No, the area of the rhombus and square are not equal. However, their area could be calculated the same way, given their measures. Area of a rhombus or any parallelogram = base × height. In a rhombus, the side and height are not the same. However, the area of the square = side×side, wherein the side could also be the height of the square. A square is a rhombus because it has four sides, and each side is the same length. However, the square is further defined as a shape that has four equal angles of 90 degrees. Therefore, a square is a rhombus. However, a rhombus is not necessarily a square. So their areas cannot be the same.

### How to Find Area of Rhombus When Side and Altitude are Given?

The area of a rhombus can be calculated when the length of the base or side and the height is given. Here, height refers to the perpendicular distance between the parallel sides, one of which we took as a base. The formula to find an area, in this case, is given as area of a rhombus = base × height sq units.

### What is the Difference Between Area of Rhombus and Square?

The area of a square is square of its side i.e., s × s, where ‘s’ is the length of the side of a square whereas the area of a rhombus is ½ × d1 × d2, where d1 and d2 are the lengths of the diagonals.

### What is the Difference Between Area of Rhombus and Rectangle?

Area of a rectangle is l × b, where ‘l’ is the length of the rectangle and ‘b’ is the breath of the rectangle whereas the area of a rhombus is ½ × d1 × d2, where d1 and d2 are the lengths of the diagonals.

### What is the Difference Between Area of Rhombus and its Perimeter?

The perimeter of a rhombus is the total measure of its boundary and it is calculated by adding the length of all its sides whereas area of a rhombus or any parallelograms is a product of its base and height, i.e., base × height. Hence the area of rhombus is half the product of ts diagonals given as ½ × d1 × d2, where d1 and d2 are the lengths of the diagonals.

### What is the Use of Area of a Rhombus Calculator?

Area of a rhombus calculator is an online tool with the help of which we can easily evaluate the area of rhombus in seconds. To determine the value we need to enter the specific parameters such as value of the diagonals. Try Cuemath’s online rhombus area calculator for quick calculations and solve your problems related to area of rhombus within seconds.

### What is the Altitude if Area of Rhombus is Given?

To calculate the altitude or height when the area is given, we require the length of the base. The formula that can be applied to calculate height is given as Area/base units.

**Properties of Rhombus**

- In a rhombus all sides are equal
- In a rhombus opposite angles are equal.
- Also, in a rhombus the sum of adjacent angles are supplementary i.e. (∠B + ∠C = 180°).
- In a rhombus, if one angle is right, then all angles are right.
- In a rhombus, each diagonal of a rhombus divides it into two congruent triangles.
- Diagonals of a rhombus bisect each other and also perpendicular to each other.

**The Perimeter of a Rhombus**

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

So, **Perimeter of rhombus = 4 × side **

**P = 4s**

Where,

s | length of a side of a rhombus |

**Derivation of Area of Rhombus**

Let ABCD is a rhombus whose base AB = b, DB ⊥ AC, DB is diagonal of rhombus = d_{₁}, AC is diagonal of rhombus = d_{₂}, and the altitude from C on AB is CE, i.e., h.

i) Area of rhombus ABCD = 2 Area of ∆ ABC

Therefore, area of rhombus = ^{1}/_{2}(product of diagonals) square units

**Solved Examples**

Q.1. What is the perimeter of a rhombus ABCD whose diagonals are 16 cm and 30 cm ?

Solution: Given d_{1} = 30 cm and d_{2} = 16 cm

=17 cm

Since, AB=BC=CD=DA,

Perimeter of ABCD = 17 × 4 = 68 cm

Q.2.Find the area of the rhombus having each side equal to 17 cm and one of its diagonals equal to 16 cm.

Solution: In rhombus ABCD, AB = BC = CD = DA = 17 cm

AC = 16 cm, AO = 8 cm

In ∆ AOD,

AD² = AO² + OD²

17² = 8² + OD²

289 = 64 + OD²

225 = OD²

OD = 15 cm

Therefore, BD = 2 OD

= 2 × 15

= 30 cm

**How to find the Area of a Rhombus?**

Area of a rhombus can be derived in 3 ways:

- Using diagonals – A = ½ × d
_{1}× d_{2}. - Using Base and Height – A = b × h.
- Using trigonometry – A = b
^{2}× Sin(a).

Where

- d
_{1}= Length of diagonal 1. - d
_{2}= Length of diagonal 2. - b = Length of any side.
- h = Height of rhombus.
- a = Measure of interior angle.

## FAQs on Area of a Rhombus Formula

**What is a parallelogram?**

In geometry, you will come across many shapes and sizes which you have to study to gain a perspective on how to calculate the area and perimeter of certain objects. The most important shapes in Euclidean geometry are the triangle, circle, and quadrilateral. In the triangle, you have an obtuse angle triangle, right-angle triangle, acute angle triangle, etc. Similarly in Quadrilaterals, when the two opposite sides are parallel to each other, it is known as a parallelogram.

**What do we mean by area and perimeter?**

The terms area and perimeter are important quantities that describe the characteristics of space occupied by a certain object. The perimeter can simply be said as the measure of the outer edge of an object which runs throughout the shape of the body and provides it with a certain shape. The area, however, does not specifically depend on boundary conditions and is measured as the space occupied by the enclosed boundaries of a body.

**Why is the importance of the rhombus?**

The rhombus is a very essential object in Euclidean geometry as it signifies one of the types of a parallelogram in which all sides are equal. Now in a body where all sides are equal, the shape is by default going to be a parallelogram but many confuse that shape with a square. A square is a certain condition of a rhombus and hence while all squares are rhombuses, all rhombuses may not necessarily be squares.

**How can the study materials along with exercise questions be downloaded from Vedantu?**

The online resources at Vedantu can easily be accessed using 4 steps:

- Open the Website of Vedantu on your Laptop or you can log in to the Vedantu App through your phone.
- Search the respective topic along with the particular exercise you are looking for.
- Click on Download PDF to download the solution in PDF format.
- Enter OTP and then the solutions will be sent to your email ID.

**I am having trouble while solving questions on rhombus, what shall I do?**

The rhombus is a very important part of Euclidean geometry and students must have good command over the topic as questions regarding this shape come regularly in school examinations and also in entrance level examinations like the JEE Mains and JEE Advanced. Try asking your doubts to your teachers when the subject is being taught in the class. You can access the online study materials at Vedantu which are designed in a student-friendly way to ensure a proper understanding of the basics of the topic and the exercises have been provided with step-by-step solutions.

### Rhombus

A rhombus is a diamond-shaped quadrilateral with equal sides but unequal angles of inclination between these two sides. It has four sides that are all the same length since it is a quadrilateral.

**Properties of a rhombus**

- All of the sides are the same length, and opposite sides are parallel to one another.
- Adjacent angles add up to 180°, but opposed angles remain constant.
- The diagonals are perpendicular to each other and bisect the angles between the sides, i.e. the vertex angles.
- The total of the angles in the Rhombus is 360°.
- If each vertex angle is 90°, the rhombus is a square.

### Formulae of Rhombus

The formulae of the rhombus include the formula for the area in different ways consisting of different formulas, the formula also includes the perimeter of the rhombus. Let’s take a look at these formulae,

**Area of Rhombus **

The entire space covered or encompassed by a rhombus on a two-dimensional plane is defined as its area. The area of a rhombus may be computed using three distinct methods: diagonal, base and height, and trigonometry.

**I**It is half of the product of the diagonal lengths.^{st}Case By using Diagonal:

Area of Rhombus = (d_{1}× d_{2})/2 sq. unitsWhere, d

_{1}is the length of diagonal 1 and d_{2}is the length of diagonal 2.

**2**The base of a rhombus is one of its sides, and the height is the perpendicular distance from the chosen base to the opposing side.^{nd}Case By using Base and Height:

Area of a Rhombus = base × height sq unitsWhere, b is the length of any side of the rhombus and h is the height of the rhombus .

**3**^{rd}case By using Trigonometry

Area of Rhombus: (side )^{2 }× sin(A) sq. unitsHere square the side of Rhombus

And Sin (A) is the interior angle.

**Perimeter of Rhombus **

A rhombus’ perimeter is the sum of its four sides or It is the product of the length of one side by 4.

Hence theperimeter of the rhombus formula = 4a, where ‘a’ is the side.

Perimeter of Rhombus = side + side + side + side = 4s.

**Rhombus Perimeter Using Diagonal Lengths**

Given a horizontal diagonal length of a and a vertical diagonal length of b, the perimeter is calculated as follows:

P = √(a^{2}+ b^{2}) × 2

### Sample Questions

**Question 1: What is the area of the rhombus for which the length of diagonals is 6 cm, 8 cm.**

**Solution:**

Given lengths of diagonals,

Diagonal (D_{1}) = 6cm

Diagonal (D_{2}) = 8cm

By using Diagonal formula : Area of rhombus = (d_{1} × d_{2})/2 sq. units

= (1/2) × 6 × 8

= (1/2) × 48

= 24

So the area of rhombus is 24 cm^{2}.

**Question 2: Calculate the area of a rhombus (using base and height) if its base is 6 cm and height is 2 cm.**

**Solution: **

Given,

Base (b) = 6 cm

height of rhombus(h) = 2 cm

Now,

Area of the rhombus(A) = base × height

= 6 × 2

= 12 cm

^{2}

**Question 3: Find the diagonal of a rhombus if its area is 120 cm ^{2 }and the length measure of the longest diagonal is 12 cm.**

**Solution:**

Given: Area of rhombus = 120 cm^{2} and Diagonal d_{1} = 12 cm.

Hence, Area of the rhombus formula, A = (d_{1} × d_{2})/2 square units, we get

120 = (12 × d_{2})/2

120 = 6 × d_{2}

Or d_{2} = 120/6

d_{2} = 20

Therefore, the Length of another diagonal is 20 cm.

**Question 4: Find the perimeter of a rhombus whose side is 7 cm.**

**Solution: **

Given side s = 7 cm

Therefore, Perimeter of Rhombus: 4 × s

So, Perimeter (P) = 4 × 7 cm = 28 cm

**Question 5: Find the side length of a rhombus whose perimeter is given as 60cm.**

**Solution: **

Given Perimeter(P) = 60 cm

Perimeter = 4 × side

Side = P/4

So, side = 60/4

= 15 cm

Hence, the length of rhombus is 15 cm.

**Question 6: Find the perimeter of the rhombus given the diagonal lengths are 3 cm and 4 cm respectively.**

**Solution:**

When diagonal lengths are Given a = 3 cm, b = 4 cm

Perimeter(P) = 2 × √(a^{2} + b^{2})

= 2 × √(3^{2} + 4^{2})

= 2 × √(9 + 16)

= 2 × 5

= 10 cm

**Question 7: Find the perimeter of a rhombus whose side is 3.5 cm.**

**Solution: **

Given that side s = 3.5 cm

Perimeter of Rhombus is given by: 4 × s

So, Perimeter (P) = 4 × (3.5) cm

= 14 cm

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