Introduction
A rhombus is a type of a parallelogram in which all the sides are congruent. This implies that a rhombus has four equal sides. The major distinction between a square and a rhombus is that each interior angle of the square is a right angle, but this is not the case in a rhombus.
As we already know, the region enclosed by a rhombus in a two-dimensional plane is known as the area of the rhombus.
A rhombus is defined by the following attributes.
- All of the sides of a rhombus are equal in length, making it an equilateral quadrilateral
- Diagonals in a rhombus intersect each other at right angles
- The diagonals are the angle bisectors in a rhombus
Formula for the Area of a Rhombus
The area of a rhombus is half the product of the lengths of the diagonals.
Depending on the characteristics we are aware of, various formulas can be employed to determine the area of a rhombus. These formulas are:
- Using the base and height,
A = b \(\times\) h
- Using the diagonal length,
A = \(\frac{1}{2}\times d_1\times d_2\)
- Using trigonometry,
A = \(b^2 \times sin(a)\)
Where,
A = Area of rhombus
\(d_1\) = Length of the first diagonal
\(d_2\) = Length of the second diagonal
b = Side length
h = Height of the rhombus
a = Measure of any interior angle
Solved Examples
Example 1: Determine the area of a rhombus with diagonals measuring 8 cm and 10 cm.
Solution:
Data stated in the question,
First diagonal, \(d_1\) = 8 cm
Second diagonal, \(d_2\) = 10 cm
A = \(\frac{d_1\times d_2}{2}\) Write the formula for the area of a rhombus
= \(\frac{8\times 10}{2}\) Substitute the values
= \(\frac{80}{2}\) Multiply
= 40 Divide
Hence, the area of the rhombus is 40 square centimeters.
Example 2: Determine the length of the second diagonal if the area of a rhombus is 168 square inches and the length of one of its diagonals is 21 inches.
Solution:
The data provided in the question,
First diagonal, \(d_1\) = 21 in
Area of rhombus, A = 168 \(\text{in}^2\)
A = \(\frac{d_1\times d_2}{2}\) Write the formula for the area of a rhombus
168 = \(\frac{21\times d_2}{2}\) Substitute the values
168 \(\times\) 2 = 21 \(\times~d_2\) Multiply each side by 2
16 = \(d_2\) Divide each side by 21
So, the length of the unknown diagonal of the rhombus is 16 inches.
Example 3: Calculate the area of a flower bed in the shape of a rhombus that has a side length of 9 feet and the perpendicular distance between the opposite sides being 7 feet.
Solution:
As stated in the question,
Side length of the flower bed, b = 9 feet
Height of the flower bed, h = 7 feet
A = b \(\times\) h Write the formula for the area of a rhombus
= 9 \(\times\) 7 Substitute the values
= 63 Multiply
So, the area of the flower bed is 63 square feet.
The area of a rhombus is the entire surface or a region enclosed by a rhombus in a two-dimensional plane. Area is always expressed in square units like square centimeters, square feet, square inches and square yards.
When the base or side lengths and the height are known, the area of a rhombus can be computed by multiplying the base with the altitude or height.
The rhombus has four sides that are congruent, but only the opposite sides are congruent in a rectangle. Each interior angle of a rectangle is a right angle but not in a rhombus.
In order to compute the altitude or height, we divide the area by the base, that is, area/base provides the altitude or height of the rhombus.