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.

These are your two diagonals.

**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 cm^{2}.

**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.

**Brush up your knowledge**

- The base of a rhombus is 5cm and its height is 7 cm. Find out its area.
- The difference between two opposite points of a rhombus is 6 cm and 8 cm respectively. Find out its area.
- 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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