What is the diagonal formula 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.

Inhaltsverzeichnis Show

What is the Area of a Rhombus?
Area of Rhombus Formula
Derivation for Rhombus Area Formula
How to Calculate Area of Rhombus?
Area of Rhombus Using Diagonals: Method 1
Area of Rhombus Using Base and Height: Method 2
Area of Rhombus Using Trigonometry: Method 3
Solved Problem on Area of Rhombus Formula
Practice Questions
More Topics Related to Rhombus Area:

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.

Read: Mathematics for grade 10

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 × d1 × d2
Using Base and Height	A = b × h
Using Trigonometry	A = b2 × Sin(a)

Where,

d1 = length of diagonal 1
d2 = 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) d1 × (½) d2 sq. units

= 4 × (1/8) d1 × d2 square units

= ½ × d1 × d2

Therefore, the Area of a Rhombus = A = ½ × d1 × d2

Where d1 and d2 are the diagonals of the rhombus.

Try This: Area of Rhombus Calculator

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. d1. 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. d2 which is the distance between B and D.

Step 3: Multiply both the diagonals, d1, and d2.

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, d1 = 6 cm

Diagonal 2, d2 = 8 cm

Area of a rhombus, A = (d1 × d2) / 2

= (6 × 8) / 2

= 48 / 2

= 24 cm2

Hence, the area of the rhombus is 24 cm2.

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 cm2

A = 70 cm2

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 = s2 × sin (30°)

A = 4 × 1/2

A = 2 cm2

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:

Area of Rhombus Example Question

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,

AD2 = AO2 + OD2

⇒ 172 = AO2 + 82

⇒ 289 = AO2 + 64

⇒ 225 = AO2

⇒ AO = 15 cm

Therefore, AC = 2 × AO

= 2 × 15

= 30 cm

Now, the area of the rhombus

= ½ × d1 × d2

= ½ × 16 × 30

= 240 cm2

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.

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.

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

A = ½ × d1 × d2

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

A = Base × Height

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

P = 4a units

If “a” be its sides and “θ” is an included angle, then the formula is:
Area of a Rhombus = a2 sin θ square units.

We know that, Area of Rhombus = (½) × Diagonal 1 × Diagonal 2 Substituting the values, we get

A = (½) × 4 × 6 = 12 cm2.

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

Assume quadrilateral

is a rhombus. If the perimeter of

and the length of diagonal

, what is the length of diagonal

Possible Answers:

Correct answer:

Explanation:

To find the value of diagonal

, we must first recognize some important properties of rhombuses. Since the perimeter is of

, and by definition a rhombus has four sides of equal length, each side length of the rhombus is equal to

. The diagonals of rhombuses also form four right triangles, with hypotenuses equal to the side length of the rhombus and legs equal to one-half the lengths of the diagonals. We can therefore use the Pythagorean Theorem to solve for one-half of the unknown diagonal:

, where

is the rhombus side length,

is one-half of the known diagonal, and

is one-half of the unknown diagonal. We can therefore solve for

is therefore equal to

. Since

represents one-half of the unknown diagonal, we need to multiply by

to find the full length of diagonal

The length of diagonal

is therefore

Assume quadrilateral

is a rhombus. If the area of

square units, and the length of diagonal

units, what is the length of diagonal

Possible Answers:

Correct answer:

Explanation:

This problem relies on the knowledge of the equation for the area of a rhombus,

, where

is the area, and

and

are the lengths of the individual diagonals. We can substitute the values that we know into the equation to obtain:

Therefore, our final answer is that the diagonal

If the area of a rhombus is

, and one of the diagonal lengths is

, what is the length of the other diagonal?

Possible Answers:

Correct answer:

Explanation:

The area of a rhombus is given below.

Substitute the given area and a diagonal. Solve for the other diagonal.

If the area of a rhombus is

, and a diagonal has a length of

, what is the length of the other diagonal?

Possible Answers:

Correct answer:

Explanation:

The area of a rhombus is given below. Plug in the area and the given diagonal. Solve for the other diagonal.

The area of a rhombus is

. The length of a diagonal is twice as long as the other diagonal. What is the length of the shorter diagonal?

Possible Answers:

Correct answer:

Explanation:

Let the shorter diagonal be

, and the longer diagonal be

. The longer dimension is twice as long as the other diagonal. Write an expression for this.

Write the area of the rhombus.

Since we are solving for the shorter diagonal, it's best to setup the equation in terms

, so that we can solve for the shorter diagonal. Plug in the area and expression to solve for

is a rhombus with side length

. Diagonal

has a length of

. Find the length of diagonal

Possible Answers:

Correct answer:

Explanation:

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle

to find the length of diagonal

. From the problem, we are given that the sides are

and

. Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

is a rhombus.

and

. Find

Possible Answers:

Correct answer:

Explanation:

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle

to find the length of diagonal

. From the problem, we are given that the sides are

and

. Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

is a rhombus.

and

. Find

Possible Answers:

Correct answer:

Explanation:

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle

to find the length of diagonal

. From the problem, we are given that the sides are

and

. Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

is a rhombus.

, and

. Find

Possible Answers:

Correct answer:

Explanation:

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle

and use the Pythagorean Theorem to solve for

. From the problem:

Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

Using the quadratic formula,

With this equation, we get two solutions:

Only the positive solution is valid for this problem.

is a rhombus.

, and

. Find

Possible Answers:

Correct answer:

Explanation:

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle

and use the Pythagorean Theorem to solve for

. From the problem:

Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

Factoring,

and

The first solution is nonsensical for this problem.

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