What is a Rhombus?
A rhombus is a 4-sided polygon that is equilateral, with four equal sides, but it’s not equiangular, and its opposite sides are parallel to each other. The diagonals of a rhombus bisect each other at right angles, and bisect a pair of opposite angles.
In our daily life, we come across multiple objects that are in the shape of rhombus such as kites, faces of diamonds or lamps,and many more.
[Note: A square is a special case of rhombus in which all angles are right angles.]
What is the Area of a Rhombus?
Just like any other 2D shape, the area of a rhombus is defined as the measure of the region enclosed within its sides. We can calculate the area of a rhombus using the measure of its diagonals.
\(Area of Rhombus, A = \frac{1}{2} \times d_{1} \times d_{2}\) square units, where \(d_{2}\text{ }and\text{ }d_{2}\) are its diagonals.
The area of any closed geometric shape is always measured in square units, so we can use units such as square millimeters, square centimeters, square meters, square inches, square feet, square yards, and square miles to measure the area of a rhombus.
Solved Examples
Example 1: Find the area of the figure.
Solution:
The given figure is a rhombus.
\(A = \frac{1}{2} \times d_{1} \times d_{2}\) [Formula for the area of a rhombus]
\(= \frac{1}{2} \times 17 \times 28\) [Substitute 17 for \(d_{1}\) and 28 for \(d_{2}\)]
\(= 238\) [Simplify]
So, the area of the figure is 238 square centimeters.
Example 2: What is the measure of the diagonal of a rhombus whose area is 144 square meters and one of its diagonals is 16 meters?
Solution:
\(A = \frac{1}{2} \times d_{1} \times d_{2}\) [Formula for the area of a rhombus]
\(144 = \frac{1}{2} \times 16 \times d_{2}\) [Substitute 144 for A and 16 for d1]
\(144 = 8 \times d_{2}\) [Simplify]
\(\frac{144}{8} = d_{2}\) [Divide both sides by 8]
\(18 = d_{2}\) [Simplify]
\(d_{2} = 18\) [Rewrite the above equation]
So, the other diagonal of the rhombus is 18 meters.
Example 3: John and Olivia are making kites on their own. Both their kites have the shape of a rhombus. The diagonals of John’s kite are 30 centimeters and 18 centimeters long, and the diagonals of Olivia’s kite are 27 centimeters and 20 centimeters long. Who makes the bigger kite?
Solution:
We can determine who makes the bigger kite by finding the area of both kites.
Length of the diagonals of John’s kite: 30 cm and 18 cm
Length of the diagonals of Olivia’s kite: 27 cm and 20 cm
\(A= \frac{1}{2} \times d_{1} \times d_{2}\) [Formula for area of rhombus]
\(Area of John’s kite = \frac{1}{2} \times 30 \times 18\) [Substitute 30 for \(d_{1}\) and 18 for \(d_{2}\)]
\(= 270\) [Simplify]
\(Area of olivia’s kite = \frac{1}{2} \times 27 \times 20\) [Substitute 27 for \(d_{1}\) and 20 for \(d_{2}\)]
\(= 270\) [Simplify]
So, the area of John’s kite is 270 square centimeters and Olivia’s is 270 square centimeters.
So, John and Olivia make kites of equal size.
A square and a rhombus are both 4-sided polygons with equal sides and with opposite sides that are parallel to each other. In a square, all the angles are right angles, but in a rhombus, all the angles may not be right angles.
A kite is a shape in which there are two pairs of adjacent sides that are of equal length. The sides adjacent to each other in a rhombus are equal, so we can say that all rhombuses are kites. However, kites may not be rhombuses.
A rhombus is not equiangular, so a square is a rhombus, but a rhombus is not a square. If the interior angles of a rhombus measure 90 degrees each only then will a rhombus can be identified as a square.
Rhombus is a quadrilateral and this means it is a 4-sided polygon. So, the sum of the interior angles of the rhombus is 360°.