The area of a rhombus refers to the space enclosed within its four equal sides. While calculating the area of common shapes like squares and rectangles is often straightforward, the area of a rhombus requires a bit more understanding of its unique properties.
Unlike squares, a rhombus does not always have right angles, which means we must rely on its diagonals, side length, or height to calculate its area.
In this blog, we will discuss the different methods to find the area of a rhombus with solved examples to help you understand the concept better.
What is a Rhombus?
A rhombus is a type of quadrilateral, which means it has four sides. It is a special kind of parallelogram where all four sides are of equal length. The opposite sides of a rhombus are parallel, meaning they are always the same distance apart and never meet, no matter how far they extend.
Additionally, the diagonals of a rhombus are unique. These are the lines that stretch from one corner to the opposite corner. The diagonals intersect at right angles (90 degrees) and bisect each other, meaning they cut each other in half.
Unlike a square, the angles in a rhombus do not have to be 90 degrees. Instead, the opposite angles are equal, making the shape distinct from other quadrilaterals.
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What is the Area of a Rhombus?
The area of a rhombus is the amount of space that is enclosed inside the shape. It tells us how much room the rhombus covers. Just like other shapes (like squares or rectangles), the area of a rhombus is measured in square units (for example, cm², m², in², etc.).
To understand the area of a rhombus better, think about it as the space inside a diamond shape because it often looks like a tilted square, which is why it is also known as a diamond-shaped quadrilateral.
Why is the Area of a Rhombus Important?
Understanding how to find the area of a rhombus is important in various fields, such as geometry, architecture, and design, because it helps you calculate how much space is available within a rhombus-shaped figure.
For example, if you're designing a tile pattern or need to measure land or floor space that has a rhombus shape, knowing the area can help you figure out how much material is needed.
The formula for finding the area of a rhombus depends on what information you know about the rhombus. In general, it involves knowing the
diagonals
(the lines that connect opposite corners) or the
base and height
of the rhombus.
The diagonals intersect at right angles (90 degrees), dividing the rhombus into four right-angled triangles, which is why the diagonals play such an important role in calculating the area.
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How to Find the Area of a Rhombus?
There are three main methods to calculate the area of a rhombus. Here's a breakdown of each approach:
Approach 1: Using the Diagonals
If you are given the lengths of the two diagonals of the rhombus, you can easily calculate the area using this formula:
Area of rhombus=1/ 2×D×d
Where:
-
D
is the length of the longer diagonal, and
-
d
is the length of the shorter diagonal.
The diagonals of a rhombus intersect at right angles, and the area is determined by multiplying the diagonals and dividing by 2. This method works because the diagonals divide the rhombus into four right-angled triangles, and the area is the sum of these triangles.
Approach 2: Using Base and Height
If you know the length of one side (which serves as the base) and the
height
(the perpendicular distance between two opposite sides), you can calculate the area as follows:
Area = Base × Height
Here:
-
The base
is the length of any side of the rhombus as all sides are equal.
-
The height
is the perpendicular distance from one side to the opposite side.
This approach is similar to how you find the area of a rectangle, but with the added consideration that the rhombus's sides are slanted.
Approach 3: Using Side Length and Angle
When you have the
length of a side
and the
measure of an interior angle
, you can use trigonometry to find the area. The formula is:
Area=Side
2
× sin(θ)
Where:
-
Side
is the length of one side of the rhombus, and
-
θ
is any interior angle (since opposite angles in a rhombus are equal, any angle will work).
The sine function is used here because the area is related to the geometry of the rhombus, and the sine of the angle gives you the height of the triangles formed by the diagonals.
The value of sine for different
an
gles is discussed below:
|
Sinθ
|
0°
|
30°
|
45°
|
60°
|
90°
|
120°
|
|
|
|
|
0
|
1/2
|
1/√2
|
√3/2
|
1
|
√3/2
|
|
|
|
Each approach is useful in different situations, and knowing these methods allows you to calculate the area of any rhombus based on the available information.
Area of a Rhombus Solved Example
1:
What is the area of a rhombus with diagonals of 8 cm and 20 cm?
Solution:
Given:
-
Length of diagonal 1 (
D
1
) = 20 cm
-
Length of diagonal 2 (
D
2
) = 8 cm
To find the area of the rhombus, we use the formula:
Area of rhombus=1/2×D
1
×D
2
Substituting the given values:
Area
=1/2 × 20 × 8 = 12 ×160 =80 cm²
So, the area of the rhombus is
80 cm²
.
2. Find the area of a rhombus where the base is 10 cm, and the height is 8 cm.
Solution:
Given:
Base length = 10 cm
Height = 8 cm
Using Approach 2 (base and height method):
Area of rhombus=10×8=80 cm²
So, the area of the rhombus is
80 cm²
.
3. Find the area of a rhombus where the side length is 8 cm and the interior angle is 60°.
Solution:
Given:
Side length = 8 cm
Interior angle = 60°
Using Approach 3 (side and angle method):
Area of rhombus=8
2
×sin(60
∘
)=64
×√3/ 2
≈
64
×
0.866
=
55.54
cm
2
So, the area of the rhombus is approximately 55.54 cm
2
.
4. Find the area of a rhombus with a side length of 9 cm and an interior angle of 120°.
Solution:
Given:
Side length = 9 cm
Interior angle = 120°
Using Approach 3 (side and angle method):
Area of rhombus=9
2
×sin(120
∘
)=81×sin(120
∘
)=81×√3/ 2≈81×0.866=70.2 cm
2
So, the area of the rhombus is approximately
70.2 cm²
.
