Area of A Rhombus: The area of a geometrical figure is the amount of space enclosed by it. A rhombus is a special type of parallelogram in which two pairs of opposite sides are congruent. The area of a rhombus depicts the total number of unit squares that can fit into it and it is measured in square units. All the students must know about different geometrical figures and the formulas for their areas. The space below has all the information about the area of a rhombus, including formulas, derivations, and more. Join Safalta School Online and prepare for Board Exams under the guidance of our expert faculty. Our online school aims to help students prepare for Board Exams by ensuring that they have conceptual clarity in all the subjects and can score their maximum in the exams.
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Area of Rhombus Formula
Different formulas to find the area of a rhombus are:
| Formulas to Calculate Area of Rhombus | |
|---|---|
| Using Diagonals | A = ½ × d1 × d2 |
| Using Base and Height | A = b × hosting |
| g Trigonometry | A = b2 × Sin(c) |
Where,
- d1 = length of diagonal 1
- d2 = length of diagonal 2
- b = length of any side
- h = height of rhombus
- c = measure of any interior angle
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How to Calculate the Area of Rhombus?
The methods to calculate the area of a rhombus are explained below with examples. There exist three methods for calculating the area of a rhombus, they are:
- Method 1: Using Diagonals
- Method 2: Using Base and Height
- Method 3: Using Trigonometry
Area of Rhombus Using Diagonals: Method 1
Consider a rhombus ABCD, having two diagonals, i.e. AC & BD.
- Step 1: Find the length of diagonal 1, i.e. d1. It is the distance between A and C. The diagonals of a rhombus are perpendicular to each other by making 4 right triangles when they intersect each other at the center of the rhombus.
- Step 2: Find the length of diagonal 2, i.e. d2 which is the distance between B and D.
- Step 3: Multiply both the diagonals, d1, and d2.
- Step 4: Divide the result by 2.
The resultant will give the area of a rhombus ABCD.
Let us understand more through an example.
Example 1: Calculate the area of a rhombus having diagonals equal to 6 cm and 8 cm.
Solution:
Given that,
Diagonal 1, d1 = 6 cm
Diagonal 2, d2 = 8 cm
Area of a rhombus, A = (d1 × d2) / 2
= (6 × 8) / 2
= 48 / 2
= 24 cm square
Hence, the area of the rhombus is 24 cm square.
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Area of Rhombus Using Base and Height: Method 2
- Step 1: Find the base and the height of the rhombus. The base of the rhombus is one of its sides, and the height is the altitude which is the perpendicular distance from the chosen base to the opposite side.
- Step 2: Multiply the base and calculated height.
Let us understand this through an example:
Example 2: Calculate the area of a rhombus if its base is 10 cm and height is 7 cm.
Solution:
Given,
Base, b = 10 cm
Height, h = 7 cm
Area, A = b × h
= 10 × 7 cm square
A = 70 cm square
Area of Rhombus Using Trigonometry: Method 3
- Step 1: Square the length of any of the sides.
- Step 2: Multiply it by Sine of one of the angles.
Let us see one example.
Example 3 Calculate the area of a rhombus if the length of its side is 2 cm and one of its angles A is 30 degrees.
Solution:
Given,
Side = s = 2 cm
Angle A = 30 degrees
Square of side = 2 × 2 = 4
Area, A = s2 × sin (30)
A = 4 × 1/2
A = 2 cm square
Practice Question Based on Area of Rhombus Formula
Question 1: Find the area of the rhombus having each side equal to 17 cm and one of its diagonals equal to 16 cm.
Solution: ABCD is a rhombus in which AB = BC = CD = DA = 17 cm
Diagonal BD = 16 cm (with O being the diagonal intersection point)
Therefore, BO = OD = 8 cm
In ∆ AOD,
AD2 = AO2 + OD2
⇒ 172 = AO2 + 82
⇒ 289 = AO2 + 64
⇒ 225 = AO2
⇒ AO = 15
Therefore, AC = 2 × AO
= 2 × 15
= 30 cm
Now, the area of the rhombus
= ½ × d1 × d2
= ½ × 16 × 30
= 240 cm square
Question 2: Examine whether the given points form a rhombus. A(2, -3), B (6, 5) C (-2, 1), and D (-6, -7)
Solution : Distance Between Two Points (x1, y1) and (x2 , y2)
√(x2 - x1)2 + (y2 - y1)2
Length of AB :
Here x1 = 2, y1 = -3, x2 = 6 and y2 = 5
= √(6-2)2+(5-(-3))2
= √42 + 82
= √(16+64)
= √80 units
Length of BC :
Here x1 = 6, y1 = 5, x2 = -2 and y2 = 1
= √(-2-6)² + (1-5)²
= √(-8)² + (-4)²
= √64 + 16
= √80 units
Length of CD :
Here x1 = -2, y1 = 1, x2 = -6 and y2 = -7
= √(-6-(-2))2+(-7-1)2
= √(-6+2)2 + (-8)2
= √(-4)2+64
= √(16+64)
= √80 units
Length of DA :
Here x1 = -6, y1 = -7, x2 = 2 and y2 = -3
= √(2-(-6))² + (-3-(-7))²
= √(2+6)² + (-3+7)²
= √(8²+4²)
= √(64+16)
= √80 units
Since all sides are equal, it may be a square also. To prove it is a rhombus, we can prove any one of the following.
The slope of the diagonal AC :
A(2, -3) C (-2, 1)
Slope = (y2-y1)/(x2-x1)
m1 = (1+3)/(-2-2)
m1 = -1
The slope of the diagonal BD :
B (6, 5) and D (-6, -7)
Slope = (y2-y1)/(x2-x1)
m2 = (-7-5)/(-6-6)
m2 = 1
Slope of AC x Slope of BD = -1(1)
= -1
The midpoint of diagonal AC :
A(2, -3) C (-2, 1)
Midpoint of AC = (x1+x2)/2, (y1+y2)/2
= (2-2)/2, (-3+1)/2
= (0, -1)
The slope of the diagonal BD :
B (6, 5) and D (-6, -7)
Midpoint of BD = (6-2)/2, (5-7)/2
= (0, -1)
Length of diagonal AC :
A(2, -3) C (-2, 1)
= √(2+2)²+(-3-1)²
= √(16+16)
= √32
The slope of the diagonal BD :
B (6, 5) and D (-6, -7)
= √(6+6)²+(5+7)²
= √(144+144)
= √288
So, the given are not vertices of the rhombus.
What is a Rhombus?
What is the Formula for Area of a Rhombus?
To calculate the area of a rhombus, the following formula is used:
A = ½ × d1 × d2
How to Find the Area of a Rhombus When the Side and Height are Given?
To find the area of a rhombus when the measures of its height and side are given, use the following formula:
A = Base × Height
What is the Formula for Perimeter of a Rhombus?
The formula to calculate the perimeter of a rhombus of side “a” is:
P = 4a units