What is a Rhombus

Rhombus

A rhombus is a convex quadrilateral with the four sides are of equal length.

Rhombus is a special type of quadrilateral.

Rhombus Formula

• perimeter of rhombus formula
• diagonal of rhombus formula
• length of sides of rhombus formula
• area of rhombus formula

Perimeter of Rhombus

Sum of the length of all sides of a rhombus is called the perimeter of rhombus.

Length of the sides are equal of a rhombus. So, one can find the perimeter of a rhombus if the length of the side is given. Again, length of side can be determined if the length of the two diagonal are given. And then one can easily evaluate the perimeter denpending on the side length.

Diagonal of Rhombus

The length of the line segment connecting two opposite vertices of a rhombus is called the diagonal of rhombus.

There are two pair of opposite vertices in a rhombus. So, the two pair of opposite vertices produce two diagonals. Hence, a rhombus contains two diagonals.

Moreover, two diagonals of a rhombus intersects at a right angle to each other.

Applying law of cosines) in △ABC,

AC² = AB² + BC² - 2AB.BC.cosB

or, d₁² = a² + a² - 2a.a cosB

or, d₁ = √2a² - 2a² cosB

or, d₁ = √2a² - 2a² cos(180°-A) [∵ A + B = 180°]

or, d₁ = √2a² - 2a² (-cosA)

or, d₁ = √2a² + 2a² cosA

or, d₁ = √a²(2 + 2cosA)

∴ d₁ = a√2 + 2cosA

Again applying law of cosines) in △ABD,

BD² = AB² + AD² - 2AB.AD.cosA

or, BD = √AB² + AD² - 2AB.AD.cosA

or, d₂ = √AB² + AD² - 2a.a cosA

or, d₂ = √a² + a² - 2a²cosA

or, d₂ = √2a² - 2a²cosA

or, d₂ = √a²(2 - 2cosA)

or, d₂ = a√2 - 2cosA

Sides of Rhombus Formula

The length of the side of rhombus can be determined if the diagonals are known. Again, the length of the side also found if area and a diagonal of a rhombus are given. Let's go how to derive a formula for finding the length of the side depending on the diagonals.

Fisrt Method: Suppose, ABCD is a rhombus of side length a whose two diagonals are d₁ ও d₂. the diagonals d₁ ও d₂ intersects at the point O. It is known that the diagonals of rhombus meet at a right angle and they bisect themselves.

∴ OA = d₁2 and OB = d₂2

∴ OA² = d₁²4 and OB² = d₂²4

Again, ∠AOB = 90°

∴ Applying pythagorean theorem from right △AOB, it can be written as

AB² = OA² + OB²

or, a² = d₁²4 + d₂²4

or, a² = d₁² + d₂²4

or, a = √d₁² + d₂²√4

∴ a = √d₁² + d₂²2

Second Method: It is known to us, sum of the area of squares drawn on the sides of a parallelogram is equal to the sum of the area of squares drawn on its two diagonals. A rhombus is also a parallelogram. Therefore, applying this on rhombus, it can be written as,

∴ AB² + BC² + CD² + AD² = d₁² + d₂²

or, a² + a² + a² + a² = d₁² + d₂²

or, 4a² = d₁² + d₂²

or, a = √d₁² + d₂²√4

∴ a = √d₁² + d₂²2

Area of Rhombus

Again, the diagonal of a rhombus can be calculated if the length of one side and its adjacent angle are given. And then, one can find the area of a rhombus depending on its diagonal. The area of a rhombus can be evaluated in several ways.

Area of Rhombus Formula

First Method: The area of a rhombus is equal to the half the product its diagonals.

Let ABCD be a rhombus and its diagonals be d₁ and d₂. If the area is A,

A = 12 d₁d₂ square unit

Second Method: It is known, the area of a parallelogram is equal to the product of its base and height. A rhombus is always a parallelogram.

So, the area of rhombus is found by product its base and height.

Let the side of a rhombus ABCD be a and its base and height be b and h respectively. If the area is A,

A = a.h square unit

Third Method: Area of a rhombus can be found by multiplying sine of any angle to the square drawn on its length of the side.

Let ABCD be a rhombus and its length of the side be a. If the area is A,

A = a²sinA square unit

If the angle is θ A = a²sinθ square unit

Fourth Method: Let a and BD be the side length and a diagonal of a rhombus ABCD respectively.

□ABCD = △ABD + △BCD

or, □ABCD = △ABD + △ABD [∵ △ABD ≅ △BCD]

or, □ABCD = 2△ABD

or, □ABCD = 2. 12 a. a sinA

∴ □ABCD = a²sinA

Properties of Rhombus

• The opposite angles of rhombus are equal to each other.
• Two diagonals are perpendicular to each other.
• All sides are equal in length.
• Two diagonals bisect the opposite angles of a rhombus.
• The two diagonals bisect each other.
• All its interior angles add up to 360°.
• Its area is half the product of two diagonals.
• The perpendicular distance between the two opposite sides is considered as the height of a rhombus.
• The two diagonals divides the rhombus into four congruent triangles.