How do you find the area of a rhombus

A rhombus has a side length of 13cm . One pair of interior angles is double the other pair of interior angles. Calculate the area of the rhombus to 2 decimal places.

Identify the length of the diagonals.

Sketching a diagram first, we have:

As the sum of angles in a rhombus is 360^{\circ} , we can quickly calculate the value of x by forming and solving an equation:

\begin{aligned} 2x+x+2x+x&=360 \\\\ 6x&=360 \\\\ x&=60^{\circ} \end{aligned}

So the angles in the rhombus are 60^{\circ} and 120^{\circ} . If we filled these in on the diagram, we have:

We can calculate the length AC by using the cosine rule c^{2}=a^{2}+b^{2}-2ab\cos(\theta) , looking at the top half of the rhombus, we have:

Labelling the sides and angles according to the formula ( \theta must be the included angle between the two sides a and b , we have:

Substituting the values of a, b, and \theta into the cosine rule, we have:

\begin{aligned} c^{2}&=a^{2}+b^{2}-2ab\cos(\theta) \\\\ c^{2}&=13^{2}+13^{2}-2\times{13}\times{13}\times\cos(120)\\\\ c^{2}&=507\\\\ c&=13\sqrt{3} \end{aligned}

So the long diagonal AC=13\sqrt{3}\text{ cm}

Repeating the same process for the diagonal BD, we have:

Substituting the values of a, b, and \theta into the cosine rule, we have:

\begin{aligned} c^{2}&=a^{2}+b^{2}-2ab\cos(\theta) \\\\ c^{2}&=13^{2}+13^{2}-2\times{13}\times{13}\times\cos(60) \\\\ c^{2}&=169 \\\\ c&=13 \end{aligned}

So the short diagonal BD=13\text{ cm} .

(For this example, BCD is an equilateral triangle!)

Write down the formula for the area of a rhombus.

A=\frac{1}{2}\times{D}\times{d}

Substitute the given values of the diagonals and solve.

As the long diagonal AC=13\sqrt{3}\text{ cm} and the short diagonal BD=13\text{ cm}, we have:

\begin{aligned} A&=\frac{1}{2}\times{D}\times{d}\\\\ &=\frac{1}{2}\times{13\sqrt{3}}\times{13}\\\\ &=146.3582932…\\\\ &=146.36\text{ cm}^2\text{ (2dp)} \end{aligned}

Write down your final answer, including the units.

The area of the rhombus ABCD is 146.36cm^2 \ (2dp). .

Area of A Rhombus: The area of a geometrical figure is the amount of space enclosed by it. A rhombus is a special type of parallelogram in which two pairs of opposite sides are congruent. The area of a rhombus depicts the total number of unit squares that can fit into it and it is measured in square units. All the students must know about different geometrical figures and the formulas for their areas. The space below has all  information about the area of a rhombus, including formulas, derivations, and more. Join Safalta School Online and prepare for Board Exams under the guidance of our expert faculty. Our online school aims to help students prepare for Board Exams by ensuring that students have conceptual clarity in all the subjects and are able to score their maximum in the exams.

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Table of content

• Area of Rhombus Formula
• How to Calculate the Area of Rhombus?
   

Area of Rhombus Formula

Different formulas to find the area of a rhombus are:

Where,

• d1 = length of diagonal 1
• d2 = length of diagonal 2
• b = length of any side
• h = height of rhombus
• c = measure of any interior angle

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• Volume Of A Sphere
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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

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.

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  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 cm square

Hence, the area of the rhombus is 24 cm square.

You may also read-

• Perimeter of a rectangle
• Area of a square
• Area of a rectangle

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 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 square

Area of Rhombus Using Trigonometry: Method 3

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

Let us see one example.

Example 3 Calculate area of a rhombus if the length of its side is 2 cm and one of its angle 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 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

Why is the area formula for a rhombus?

Why is the area of a rhombus divided by 2?