Area of Rhombus
Area of rhombus is defined as the amount of space enclosed or encompassed by a rhombus in a two-dimensional plane. Rhombus is a special type of parallelogram, having all sides equal to each other. A rhombus can be differentiated from a square through the measure of its internal angle. The internal angle of a rhombus need not be a right-angle specifically. The area of a rhombus can be calculated in different ways, depending on the parameters known to us.
|1.||What is Area of Rhombus?|
|2.||Area of a Rhombus Formula|
|3.||How to Calculate the Area of a Rhombus?|
|4.||FAQs on Area of Rhombus|
What is 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. It depicts the total number of unit squares that can fit into it and it is measured in square units (like cm2, m2, in2, etc). Rhombus is a parallelogram with the opposite sides parallel, the opposite angles equal, and the adjacent angles supplementary. The following properties are used to define a rhombus.
- A rhombus is an equilateral quadrilateral because all the sides have equal lengths.
- In a rhombus, diagonals bisect each other at right angles.
- The diagonals are angle bisectors.
- The area of a rhombus can be found in different ways: using base and height, using diagonals, and using trigonometry.
Area of a Rhombus Formula
Different formulas can be used to calculate the area of a rhombus depending upon the parameters known to us. The different formulas followed for the calculation of the area of a rhombus are,
- Using base and height
- Using diagonals
- Using trigonometry
Formula for Area of Rhombus When Base and Height Are Known
Rhombus is a parallelogram. We know that the area of a parallelogram is given by multiplying base and height sq units. The same is applied to the rhombus as well.
Area of a Rhombus = base × height sq units
Example: Find the area of a rhombus having a side length of 7 inches and the height of the rhombus of 10 inches.
Solution: As we know, Area = base × height units2
⇒ Area = 7 × 10 inches2
⇒ Area = 70 inches2
Formula for Area of Rhombus When Diagonals Are Known
The area of a rhombus is equal to half the product of the lengths of the diagonals. The formula to calculate the area of a rhombus using diagonals is given as,
Area = (\(d_1\) × \(d_2\))/2 sq. units, where, \(d_1\) and \(d_2\) are the diagonals of the rhombus.
Consider the Rhombus ABCD. Let E be the point of intersection of two diagonals. We make the following observations,
- Four sides are congruent.
- The diagonals bisect each other. ⇒ BD = BE + ED& and AC = AE + EC
- Four interior angles with opposite angles equal. ⇒ ∠ A = ∠C and ∠B = ∠D
- The two diagonals are AC and BD.
The area of the rhombus ABCD = Area of ∆ ADC + Area of ∆ ABC
Area of the rhombus = 2 × Area of ∆ ABC ---(1) (∵ ∆ ABC congruent to ∆ ADC)
Area of ∆ ABC
= 1/2 × Base × Height
= 1/2 × AC × BE
= 1/2 × AC × 1/2 × BD (∵BE = BD/2)
= 1/4 (AC × BD) --- (2)
Area of the Rhombus ABCD
Area = 2 × 1/4 × AC × BD = 1/2 × AC × BD (From (1) and (2))
⇒ Area = 1/2 × diagonal 1 × diagonal 2
∴ Area of a Rhombus = 1/2 × diagonal 1 × diagonal 2 units2
Formula for Area of Rhombus When Side and Angles Are Known
We apply the concept of trigonometry while calculating the area when sides and angles are known. We can use any angle because either the angles are equal or they are supplementary, and supplementary angles have the same sine. Area of a rhombus using side and angle is given as,
Area of a Rhombus = side2 × sin(A) sq. units, where 'A' is an interior angle.
Example: What is the area of a rhombus if the length of its side is 4 yards and one of its angle A is 30º.
Solution: As we know, area of rhombus = s2 × sin(30º)
Area of rhombus = s2 × sin(30º) = 42 × 1/2
⇒ Area of rhombus = 16 × 1/2 = 8 sq yards
How to Calculate the Area of a Rhombus?
The different methods to calculate the area of a rhombus are explained below. There are three methods for calculating the area of a rhombus, given as:
- Method 1: Using Base and Height
- Method 2: Using Diagonals
- Method 3: Using Trigonometry
Area of Rhombus Using Base and Height
- Step 1: Find and note the base and the height of the given rhombus. The base is one of its sides of a rhombus, while the height is the perpendicular distance from the chosen base to the opposite side.
- Step 2: Multiply the base and height.
The resultant value will give the area of a rhombus.
Area of Rhombus Using Diagonals
Consider a rhombus ABCD, having two diagonals, i.e. AC & BD.
- Step 1: Find the length of both the diagonals, diagonal 1 and diagonal 2.
- Step 2: Multiply both the lengths, d1, and d2.
- Step 3: Divide the result by 2.
The resultant value will give the area of a rhombus ABCD.
Area of Rhombus Using Trigonometry
- Step 1: Square the length of any of the sides.
- Step 2: Multiply it by the sine of any one of the angles.
The resultant value will give the area of a rhombus.
Example: Consider the rhombus ABCD. AB, BC, CD, DA are the congruent (equal) sides. AC and BD are the diagonals, and they meet at E. Given CD = 17 feet and AE = 8 feet.
Now we know diagonal 1, AC = 16 feet.
Next, we need to calculate BD.
BD = BE+ ED = 2 × BE
We still have an unknown, BE.
Pythagorean theorem states that
BC2 = BE2 + EC2
BC = 17 feet (∵ CD =BC as all the sides are congruent)
EC = 8 feet (∵ AE = EC as the diagonals bisect each other)
172 = BE2+ 82
⇒ BE2 = 289 - 64
∴ BE = 15 feet and BD = 30 feet
It's time to substitute all the values in the area of the rhombus formula.
Area of the Rhombus = 1/2 × \(d_1\) × \(d_2\) sq units
= 1/2 × BD × AC sq feet
= 1/2 × 16 × 30 sq feet
⇒ Area of rhombus = 240 sq feet
Tips and Tricks:
- Remember that the height is not the same as the length of the side of a rhombus.
- The area of a rhombus can be found in three ways: when diagonals are given, when an angle and a side are given, when angle and height are given.
- Use the Pythagorean theorem to find the second diagonal if the measures of one diagonal and side are given.
Solved Examples on Area of Rhombus
Example1: Using the area of a rhombus formula, find the area of the rhombus given in the figure below.
Area of a rhombus= 1/2 × BD × AC
BD = 2 × BE
= 2 × 8
= 16 yards
AC = 2 × AE = 2 × 10 = 2 yards
⇒ Area = 1/2 × 16 × 20
= 8 × 20
= 160 yards2
Answer: Area of the rhombus = 160 yards2
Example 2: The side of a rhombus measures 5 inches, while the length of its one diagonal is 8 in. Calculate its area.
Given a side and a diagonal, we will find the other diagonal.
Area = (AC × BD)/2 sq inches
⇒ AC = 8 in
⇒ AO = 4 in (∵ AO = 1/2 AC)
To find the other diagonal BD, consider AOD.
By Pythagorean theorem, AD2 = AO2 + OD
⇒ 252 = 42 + OD2
⇒ OD2 = 25 −16
⇒ OD2 = 9
⇒ OD = 3 in
⇒ BD = 6 in (∵BD = 2 × OD)
Area = (8 × 6) ÷ 2 sq inches
Area = 24 sq inches
Answer: Area of the rhombus = 24 sq inches
FAQs on Area of Rhombus
What is the Area of Rhombus?
The area of a rhombus is the total amount of space enclosed or encompassed by a rhombus in a two-dimensional plane. It is expressed in square units(like cm2, m2, in2, etc).
What is the Formula of Finding the Area of a Rhombus?
Different formulas can be used to calculate the area of a rhombus depending upon the parameters known to us. Using base and altitude the formula is given as, Area of a Rhombus = base × height sq units. Area of rhombus using diagonals is: Area = (\(d_1\) × \(d_2\))/2 sq. units, where \(d_1\) and \(d_2\) are the diagonals of the rhombus. Applying the concept of trigonometry using side and angle, we can follow the formula: Area of a Rhombus = side2 × sin(A) sq. units, where 'a' is an interior angle.
How do you Find the Side of a Rhombus With the Diagonals?
The area of a rhombus can be calculated using the lengths of diagonals. The formula to find area in this case is given as: Area = (\(d_1\) × \(d_2\))/2 sq. units, where \(d_1\) and \(d_2\) are the diagonals of the rhombus.
Are the Area of Rhombus and Square Equal?
No, the area of the rhombus and square are not equal. However, their area could be calculated the same way, given their measures. Area of a rhombus or any parallelogram = base × height. In a rhombus, the side and height are not the same. However, the area of the square = side×side, wherein the side could also be the height of the square. A square is a rhombus because it has four sides, and each side is the same length. However, the square is further defined as a shape that has four equal angles of 90 degrees. Therefore, a square is a rhombus. However, a rhombus is not necessarily a square. So their areas cannot be the same.
How to Find Area of Rhombus When Side and Altitude are Given?
The area of a rhombus can be calculated when the length of the base or side and the height is given. Here, height refers to the perpendicular distance between the parallel sides, one of which we took as a base. The formula to find an area, in this case, is given as area of a rhombus = base × height sq units.
What is the Altitude of a Rhombus Given the Area?
To calculate the altitude or height when the area is given, we require the length of the base. The formula that can be applied to calculate height is given as Area/base units.