In geometry, a rhombus is a special type of parallelogram in which two pairs of opposite sides are congruent. That means all the sides of a rhombus are equal. Students often get confused with square and rhombus. The main difference between a square and a rhombus is that all the internal angles of a square are right angles, whereas they are not right angles for a rhombus. In this article, you will learn how to find the area of a rhombus using various parameters such as diagonals, side & height, and side and internal angle, along with solved examples in each case.
What is the Area of a Rhombus?
The area of a rhombus can be defined as the amount of space enclosed by a rhombus in a two-dimensional space. To recall, a rhombus is a type of quadrilateral projected on a two dimensional (2D) plane, having four sides that are equal in length and are congruent.
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Area of Rhombus Formula
Different formulas to find the area of a rhombus are tabulated below:
| Using Diagonals | A = ½ × d1 × d2 |
| Using Base and Height | A = b × h |
| 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 any interior angle
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
Therefore, the Area of a Rhombus = A = ½ × d1 × d2
Where d1 and d2 are the diagonals of the rhombus.
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How to Calculate 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 (i.e., using side and angle)
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 centre 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 cm2
Hence, the area of the rhombus is 24 cm2.
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 the 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 cm2
A = 70 cm2
Area of Rhombus Using Trigonometry: Method 3
This method is used to calculate the area of the rhombus when the side and one of its internal angles are given.
- Step 1: Square the length of any of the sides.
- Step 2: Multiply it by Sine of one of the angles.
Let us see how to find the area of a rhombus using the side and angle in the below 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 cm2
Solved Problem on Area of Rhombus Formula
Question: Find the area of the rhombus having each side equal to 17 cm and one of its diagonals equal to 16 cm.
Solution:
Area of Rhombus Example Question
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 cm
Therefore, AC = 2 × AO
= 2 × 15
= 30 cm
Now, the area of the rhombus
= ½ × d1 × d2
= ½ × 16 × 30
= 240 cm2
Practice Questions
- Find the height of the rhombus, whose area is 175 cm² and perimeter is 100 cm.
- Calculate the area of a rhombus with a side of 5 cm, and one of the internal angles is 120 degrees.
- If the area of a rhombus is 143 sq. units and one of its diagonal is 26 units, find the other diagonal.
More Topics Related to Rhombus Area:
A rhombus is a type of quadrilateral whose opposite sides are parallel and equal. Also, the opposite angles of a rhombus are equal and the diagonals bisect each other at right angles.
To calculate the area of a rhombus, the following formula is used:
A = ½ × d1 × d2
To find the area of a rhombus when the measures of its height and side are given, use the following formula:
A = Base × Height
The formula to calculate the perimeter of a rhombus of side “a” is:
P = 4a units
If “a” be its sides and “θ” is an included angle, then the formula is:
Area of a Rhombus = a2 sin θ square units.
We know that, Area of Rhombus = (½) × Diagonal 1 × Diagonal 2 Substituting the values, we get
A = (½) × 4 × 6 = 12 cm2.
No, the area of a rhombus is not the same as the area of a square.
The area of a square is the square of its side, whereas the area of a rhombus is the half the product of diagonal 1 and diagonal 2.
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Suppose, the values of any one of the angles and the side of a rhombus are given. How to find the length of any of the diagonals?
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Find the length of a side of a rhombus that has diagonals with lengths of
Possible Answers:
Correct answer:
Explanation:
Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.
Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.
First, find the lengths of half of each diagonal.
Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.
Plug in the lengths of the half diagonals to find the length of the rhombus.
Make sure to round to
Find the length of a side of a rhombus that has diagonals with lengths of
Possible Answers:
Correct answer:
Explanation:
Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.
Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.
First, find the lengths of half of each diagonal.
Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.
Plug in the lengths of the half diagonals to find the length of the rhombus.
Make sure to round to
Find the length of a side of a rhombus that has diagonals with lengths of
Possible Answers:
Correct answer:
Explanation:
Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.
Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.
First, find the lengths of half of each diagonal.
Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.
Plug in the lengths of the half diagonals to find the length of the rhombus.
Make sure to round to
Find the length of a side of a rhombus that has diagonals with lengths of
Possible Answers:
Correct answer:
Explanation:
Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.
Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.
First, find the lengths of half of each diagonal.
Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.
Plug in the lengths of the half diagonals to find the length of the rhombus.
Make sure to round to
Find the length of a side of a rhombus that has diagonals with lengths of
Possible Answers:
Correct answer:
Explanation:
Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.
Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.
First, find the lengths of half of each diagonal.
Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.
Plug in the lengths of the half diagonals to find the length of the rhombus.
Make sure to round to
Find the length of a side of a rhombus that has diagonals with side lengths of
Possible Answers:
Correct answer:
Explanation:
Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.
Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.
First, find the lengths of half of each diagonal.
Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.
Plug in the lengths of the half diagonals to find the length of the rhombus.
Make sure to round to
Given: Parallelogram
True or false: Parallelogram must be a rhombus.
Possible Answers:
Explanation:
A rhombus is defined to be a quadrilateral with four congruent sides.
Parallelogram gives the lengths of two of its opposite sides to be congruent, but this is characteristic of all parallelograms. No information is given about the other two sides, so the figure need not be a rhombus.
Given: Parallelogram such that
True or false: Parallelogram must be a rhombus.
Possible Answers:
Explanation:
Opposite sides of a parallelogram are congruent, so
and
All four sides are congruent to one another. It follows by definition that Parallelogram is a rhombus.
Given: Quadrilateral
True or false: From the information given, it follows that Quadrilateral
Possible Answers:
Explanation:
Below are two quadrilaterals marked
The quadrilateral on the left has four congruent sides and is by definition a rhombus. The quadrilateral on the right is not a rhombus, since not all four sides are congruent.
In both cases,
Therefore, Quadrilateral
Given:
True or false: It follows from the given information that
Possible Answers:
Explanation:
As we are establishing whether or not
If we examine the sides and the angle of in the congruence statements -
The only congruence postulate dealing with two sides and an angle is the SAS Congruence Postulate, which requires congruence between two sides and the included angle of both triangles. This theorem does not apply, so we cannot prove the triangles congruent.
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