In geometry, The Rhombus formulas are used to find the formulas which are associated with a rhombus. All the sides of a rhombus are equal so it is also called an equilateral quadrilateral. The Area of rhombus is described as two-dimensional space which is enclosed by the rhombus. Also Mostly Students are often to get confused between square and rhombus. The main difference between a square and a rhombus is that a square has four equal sides and four right angles, while a rhombus has four equal sides but its angles are not necessarily right angles. In this article, you will learn all about area of rhombus using diffrent parameters such as diagonals, side & height, and side and internal angle, along with a few solved examples in the end.
What is the Area of a Rhombus?
The area of a rhombus is the measure of the surface enclosed by the four sides of a rhombus shape. It can be found by multiplying the lengths of its diagonals and dividing the result by two or by multiplying the length of any one of its sides with the perpendicular distance between opposite sides. The area of a rhombus is expressed in square units.
Area of Rhombus Formula
Rhombus formulas are defined for two features, an area of the rhombus and the perimeter of a rhombus. The Rhombus formula can be expressed as:
| Formulas to Calculate Area of Rhombus | |
|---|---|
| Using Diagonals | A = ½ × d1 × d2 |
| Using Trigonometry | A = b2 × Sin(a) |
| Using Base and Height | A = b × h |
Where,
- b = length of any side
- h = height of rhombus
- a = measure of any interior angle
- d1 = length of diagonal 1
- d2 = length of diagonal 2
Derivation for Rhombus Area Formula
Consider the following rhombus: ABCD
Let O be the point of intersection of two diagonals AC and BD.
The area of the rhombus will be:
A = 4 × area of ∆ AOB
= 4 × (½) × AO × OB sq. units
= 4 × (½) × (½) d1 × (½) d2 sq. units
= 4 × (1/8) d1 × d2 square units
= ½ × d1 × d2
Thus, Area of a Rhombus = A = ½ × d1 × d2
Where d1 and d2 are the diagonals of the rhombus.
Also Check : Important Maths Formulas for Class 8
How to Calculate Area of Rhombus?
There are three methods exist to calculate the area of a rhombus, they are:
Method 1: Using Diagonals – A = ½ × d1 × d2.
Method 2: Using Base and Height – A = b × h.
Method 3: Using Trigonometry– A = b2 × Sin(a).
Where,
d1 = Length of diagonal 1.
d2 = Length of diagonal 2.
b = Length of any side.
h = Height of rhombus.
a = Measure of interior angle.
Let us understand the area of rhombus formula better using a some solved examples.
Method 1: Area of Rhombus Using Diagonals
Examples on Area of Rhombus Using Diagonals
Example 1: Find the area of the rhombus of diagonal lengths 12 and 10 units using the rhombus formula.
Solution:
To find the area of the rhombus using the diagonal lengths of 12 and 10 units, we can use the formula:
Area = (d1 x d2)/2
where d1 and d2 are the lengths of the diagonals.
Substituting the values we have, we get:
Area = (12 units x 10 units)/2
Simplifying this, we get:
Area = 60 square units
Therefore, the area of the rhombus is 60 square units.
Method 2: Area of Rhombus Using Base and Height
Examples on Area of Rhombus Using Base and Height
Example 2: Calculate the area of a rhombus if its base is 9 cm and height is 5 cm.
Solution:
Given,
Base, b = 9 cm
Height, h = 5 cm
Area, A = b × h
= 9 × 5 cm2
A = 45 cm2
Method 3: Area of Rhombus Using Trigonometry
Examples on Area of Rhombus Using Trigonometry
Example 3: Find the area of a rhombus with diagonals of length 12 cm and 16 cm, using trigonometry.
Solution:
Using trigonometry, we can find the length of one of the sides of the rhombus as follows:
In right triangle AOB, we have:
AB^2 = AO^2 + BO^2 (by Pythagorean theorem)
AB^2 = (1/2 AC)^2 + (1/2 BD)^2 (substituting AO = 1/2 AC and BO = 1/2 BD)
AB^2 = (1/2)^2 (AC^2 + BD^2)
AB^2 = (1/2)^2 (12^2 + 16^2)
AB^2 = 200
AB = √200 = 10√2
Area of the rhombus found using the formula:
Area = (diagonal1 x diagonal2)/2
Area = (12 cm x 16 cm)/2
Area = 96 cm^2
Therefore, the area of the rhombus is 96 square cm.
Example 4: Find the area of a rhombus with side length 5 cm and an included angle of 60 degrees, using trigonometry.
Solution :
Let the side length of the rhombus be AB = 5 cm. and angle BAC be 60 degrees, and O be the intersection point of the diagonals AC and BD.
Using trigonometry, we can find the length of the diagonals as follows:
In triangle ABO, we have:
sin 60° = AO / AB
AO = AB sin 60°
AO = 5 cm x √3/2
AO = (5√3)/2
In right triangle AOC, we have:
cos 60° = AO / AC
AC = AO / cos 60°
AC = [(5√3)/2] / (1/2)
AC = 5√3
Similarly, in right triangle BOD, we have:
BD = 2 x BO = 2 x AO = 2 x (5√3)/2 = 5√3
Area of the rhombus found using the formula:
Area = (diagonal1 x diagonal2)/2
Area = (5√3 x 5√3)/2
Area = (25 x 3)/2
Area = 37.5 square cm
Therefore, the area of the rhombus is 37.5 square cm.
Also Check : Trigonometry Table
Area of a Rhombus Calculator
Area of Rhombus Formula When Side and Diagonal Is Given
formula for the area of the rhombus to get the formula given at the beginning of this answer.
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