Area of Rhombus: Meaning, Formula, and Solved Examples
The area of a rhombus is the amount of space enclosed by its four equal-length sides. Learn three different formulas to calculate the area of a rhombus with examples here.
Chandni 15 Dec, 2024
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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.
[video width="1920" height="1080" mp4="https://www.pw.live/exams/wp-content/uploads/2024/12/Curious-Jr-Ad-3-1-1-1.mp4"][/video]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. [video width="1920" height="1080" mp4="https://www.pw.live/exams/wp-content/uploads/2024/12/curious-jr.mp4"][/video]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.
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.
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).
| Sinθ | 0° | 30° | 45° | 60° | 90° | 120° | |||
| 0 | 1/2 | 1/√2 | √3/2 | 1 | √3/2 |
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
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 cmUsing 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 .| Related Articles | |
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| Simple Interest | Surface Area of Cylinder |
Area of Rhombus FAQs
How does the shape of a rhombus affect its area?
The area of a rhombus is influenced by the length of the diagonals and the angle between the sides. As the angle approaches 90° (becoming a square), the area increases for a fixed side length. Larger diagonals also increase the area.
How does the height of a rhombus help in finding its area?
The height of a rhombus is the perpendicular distance between two opposite sides. It helps in determining how much space is enclosed by the shape, particularly when combined with the length of the sides.
What makes the area of a rhombus different from a square or rectangle?
While the area of a square or rectangle is based on length and width, the area of a rhombus depends on its diagonals or side lengths and angles, which makes it more versatile in shape.
What happens to the area of a rhombus if the diagonals are longer?
As the diagonals become longer, the area of the rhombus increases. The diagonals play a direct role in determining the size of the shape.
How does the angle of a rhombus affect its area?
The angle between the sides influences the overall area. Larger angles typically result in a larger area, as they allow the rhombus to cover more space with the same side length.
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