 A rhombus is a Euclidian concept with four equal sides and two sets of angles. It is not the same as a square. The area of a rhombus can be found by various methods. Three of them have been listed here. A bit of history, definitions and differences, methods and questions have also been provided.

## Area of a Rhombus

### History first!

Today, we are going to learn about the area of a rhombus. And for that, we need to journey back in space and time to a world very different from ours. Here is a little bit of history, then. How many of you are familiar with Euclid? Euclid was a mathematician who lived in ancient Greece approximately three centuries before the birth of Christ.  Euclid’s book The Elements gives us Euclidian geometry which deals with shapes like the triangle, the parallelogram and the rhombus.

There are forms of non-Euclidian geometry too. Hyperbolic geometry and spherical geometry are two such well-known forms.

### Definitions

What is a Rhombus? • A rhombus is a quadrilateral of equal sides. Opposite sides of the quadrilateral are parallel to each other in a rhombus. Since it has parallel sides, it is also a parallelogram.
• To quickly remind you, a quadrilateral is a shape with four definite sides. The sides of a quadrilateral are not equal to each other.
• Parallel lines exist in a plane and can never meet. The opposite of parallel lines is intersecting lines.

A rhombus can easily be divided into two equal triangles. Try it!

I can already guess your first question. How does a rhombus differ from a square?

Well, the angles of a square are always 90°. A square has four right angles. It is not so in the case of rhombuses. A rhombus has two acute and two obtuse angles. Opposite sides of a rhombus are endowed with equal angles.

### Similarity: The sum of the angles of both the square and rhombus is 360°

There are two other properties of a rhombus you must know about.

• The diagonals of a rhombus bisect the angles of the rhombus. This means that the diagonal cuts the angle into two equal halves.
• The diagonals of a rhombus are perpendicular bisectors of each other. This means that the diagonals of a rhombus cut each other at 90°.

### Now, let us recapitulate what we learnt about Area of a Rhombus

The area of a shape is the amount of space enclosed within the lines of that shape. In the case of the rhombus, the area of a rhombus is what is enclosed by its four lines.

The simplest way to find an area is that of a rectangle. In the case of a rectangle, you can simply multiply its length and breadth and present your value in square units.

However, this rule is not applicable in the case of a rhombus.

#### Method 1

The basic formula for area of a rhombus is pq/2; where p and q are the two diagonals drawn across the rhombus.

In order to find the area of a rhombus, therefore,

Step 1: Draw a line connecting opposite sides of a rhombus. Connect the sides with the similar angles. This will cover only two angles.

Step 2: Now draw another line connecting opposite sides of the rhombus, as described in Step 1.

Step 3: With your ruler, measure the length of these two lines.

Step 4: In order to find the area, multiply the areas and then divide your answer by 2.

Step 5: Remember that your area will always be in square units.

#### Method 2

Here is another way to find the area of a rhombus. Let us call this the base and height method. This is even simpler than the first method.

Step 1: Find the base of your rhombus. The base of the rhombus should be your line towards the bottom. Measure this line with the help of your ruler.

Step 2: Now, from this line, draw a line that touches the top of the rhombus. This line is the height of the rhombus. Measure this line too.

Step 3: Multiply the base of the rhombus with the height of the rhombus. The value in square units is the area of the rhombus.

#### Method 3

There is a third way to find the area of a rhombus. This way involves trigonometry.

Step 1: Find the length of any of the sides of your rhombus. Any side should do as a rhombus has equal sides.

Step 2: Find the square of this length. For example, if the length is 4 cm, then the square is 4×4=16 cm2.

Step 3: Find out the angle of your rhombus. You can choose either angle. You can measure the angle with the help of a protractor.

Step 4: Once you have done that, find out the sine value of this angle. For example, if the angle is 36®, the sine value or sin 36 is 0.5877 (approx.).

Step 5: Now, multiply, the side of the rhombus with this value. The resultant value is your area of the rhombus.

Step 6: Always remember to put the area in square units.

Now that you have learnt to find the area of a rhombus, you should be able to draw them and solve sums about them.

1. The base of a rhombus is 5cm and its height is 7 cm. Find out its area.
2. The difference between two opposite points of a rhombus is 6 cm and 8 cm respectively. Find out its area.
3. The angle of a rhombus is 45°. The side of a rhombus is 6 cm. What is the area of the rhombus?

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