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Rhombus Formula: 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.
Rhombus Formula
A rhombus is a type of quadrilateral, which is a four-sided polygon. It is characterized by having four sides of equal length. In addition to having equal sides, a rhombus also has opposite sides that are parallel to each other. Furthermore, its opposite angles are congruent, meaning they have the same measure. The diagonals of a rhombus bisect each other at right angles, forming four congruent right triangles within the shape. Overall, a rhombus possesses symmetrical properties and has an appearance similar to a diamond or a square that is tilted.
The formula for a rhombus involves its side length and one of its angles. Here are some key aspects of the rhombus formula:
Area: The formula to calculate the area of a rhombus is given by A = (d₁ * d₂) / 2, where d₁ and d₂ are the lengths of the diagonals. This formula can be derived from the fact that the diagonals of a rhombus bisect each other at right angles, forming four congruent right triangles.
Side length: If you know the length of one side of a rhombus, you can find the length of the other sides using the formula s = d * sin(θ), where s represents the side length, d is the length of the diagonal, and θ is one of the angles between the diagonals.
Perimeter: The perimeter of a rhombus is simply the sum of its four side lengths. If you know the length of one side, you can find the perimeter by multiplying it by 4.
Remember, these formulas apply specifically to rhombuses, which are quadrilaterals with four equal sides.
Solved Examples on Rhombus Formula
Example 1: Given a rhombus with diagonals measuring 10 cm and 12 cm, calculate its area. Solution: Using the formula for the area of a rhombus,
A = (d₁ x d₂) / 2,
where d₁ and d₂ are the lengths of the diagonals:
A = (10 cm x 12 cm) / 2
A = 120 cm²
Therefore, the area of the rhombus is 120 square centimeters.
Example 2: In a rhombus, one of the angles between the diagonals measures 60 degrees, and the length of one side is 8 cm. Determine the lengths of the other sides.
Solution:
Using the formula for the side length of a rhombus, s = d x sin(θ),
where s is the side length, d is the length of the diagonal, and θ is the angle between the diagonals:
s = 8 cm x sin(60°)
s = 8 cm x √(3/2)
s ≈ 6.93 cm
Hence, the lengths of the other sides of the rhombus are approximately 6.93 centimeters.
These examples demonstrate the application of the rhombus formulas to find the area and side lengths of a given rhombus based on the given information.
Example 3: A rhombus has a side length of 6 cm and one of its angles measures 60 degrees. Calculate the area of the rhombus.
Solution:
To find the area of the rhombus, we can use the formula
A = (d₁ x d₂) / 2,
where d₁ and d₂ are the lengths of the diagonals.
Since the diagonals of a rhombus bisect each other at right angles, we can determine the length of one diagonal using the side length and angle.
Using the formula for the diagonal, d = s / sin(θ), where s is the side length and θ is the angle between the diagonals:
d = 6 cm / sin(60°)
d = 6 cm / (√3/2)
d = 12 cm / √3
Now we can calculate the area:
A = (12 cm x d₂) / 2
A = (12 cm x 12 cm / √3) / 2
A = (144 cm² / √3) / 2
A ≈ 83.14 cm²
Therefore, the area of the rhombus is approximately 83.14 square centimeters.
Frequently Asked Questions (FAQs) on Rhombus Formula
What are the 4 properties of a rhombus?
Answer: The four properties of a rhombus are:
Equal sides: A rhombus has four sides of equal length. This property distinguishes it from other quadrilaterals.
Opposite angles are congruent: The opposite angles of a rhombus have the same measure. In other words, if angle A is congruent to angle C, then angle B is congruent to angle D.
Diagonals bisect each other: The diagonals of a rhombus intersect each other at a 90-degree angle, bisecting each other. This means that the point of intersection divides each diagonal into two equal parts.
Diagonals are perpendicular: The diagonals of a rhombus are perpendicular to each other. This property ensures that the rhombus has four right triangles formed by its diagonals.
Yes, the diagonals of a rhombus are equal in length. This is one of the defining properties of a rhombus. Both diagonals of a rhombus intersect each other at right angles and bisect each other. Consequently, the two diagonals are of equal length, which means that the distance from one corner of the rhombus to the opposite corner is the same as the distance from another corner to its opposite corner.
No, not all angles in a rhombus are equal. While a rhombus has four sides of equal length, its angles are not necessarily equal except in special cases. The opposite angles in a rhombus are congruent, meaning they have the same measure. So, if angle A is congruent to angle C, then angle B is congruent to angle D. However, the adjacent angles within a rhombus can have different measures unless it is a special case where all angles are equal, such as a square, which is a specific type of rhombus with four right angles.
The altitude of a rhombus is the perpendicular distance between any two parallel sides of the rhombus. It is the distance from one side to the opposite side, measured along a perpendicular line. Since a rhombus has two pairs of parallel sides, it can have two altitudes—one for each pair of parallel sides. The altitudes of a rhombus intersect at a right angle at the rhombus's diagonals' point of intersection. The length of the altitude can be calculated using trigonometry or by applying the Pythagorean theorem in conjunction with the side lengths or diagonal lengths of the rhombus.
No, rhombuses do not have four right angles. A rhombus is a quadrilateral with four sides of equal length. While the opposite angles of a rhombus are congruent (meaning they have the same measure), they are not necessarily right angles unless the rhombus is a special case known as a square. A square is a type of rhombus where all four angles are right angles, making it a quadrilateral with four right angles and four equal sides. In a general rhombus, the angles can vary, but the opposite angles will always have equal measures.
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 find the area of a rhombus when the measures of its height and side are given, use the following formula: A = Base × Height
If a be its sides and θ is an included angle, then the formula is: Area of a Rhombus = a2 sin θ square units.
No, the area of a rhombus is not the same as the area of a square. Are rhombus diagonals equal?
Are all angles equal in rhombus?
What is the altitude of rhombus?
Do rhombuses have 4 right angles?
What is a Rhombus?
How to Find the Area of a Rhombus When the Side and Height are Given?
How to find the area of a rhombus if one of its sides and an included angle are given?
Is the area of a rhombus the same as the area of a square?
