How to find the area of a rhombus with vertices ?

Here we are going to see how to find the area of a rhombus with vertices.

Area of rhombus = (1/2) [d_{1} x d_{2}]

From the given vertices, we need to find the length of diagonals by using the formula distance between two points.

Once we find out the length of diagonals, we may apply those values in the above formula in order to get area of rhombus.

Let us look into some example problems to understand the above concept.

**Question 1 :**

Find the area of a rhombus if its vertices are (3,0) (4,5) (-1,4) and (-2,-1) taken in order.[Hint:Area of rhombus = (1/2) product of its diagonals]

**Solution :**

Let A(3,0) B(4,5) C(-1,4) and D(-2,-1) are the vertices of the rhombus.

length of diagonal AC = √(x₂ - x₁)² + (y₂ - y₁)²

x₁ = 3 y₁ = 0 x₂ =-1 y₂ = 4

= √(-1 - 3)² + (4 - 0)²

= √(-4)² + (4)²

= √16 + 16

= √32

length of diagonal BD = √(x₂ - x₁)² + (y₂ - y₁)²

x₁ = 4 y₁ = 5 x₂ =-2 y₂ = -1

= √(-2-4)² + (-1-5)²

= √(-6)² + (-6)²

= √36 + 36

= √72

Area of rhombus = (1/2) x √32 x √72

= (1/2)4√2 x 6√2

= 24 square units.

**Question 2 :**

Find the area of a rhombus if its vertices are A (1,4) and B (5,1) and C (1, -2) and D (-3,1)taken in order.

**Solution :**

Length of diagonal AC :

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 1, y₁ = 4, x₂ = 1 and y₂ = -2**

**= ** √(1-1)² + (-2-4)²

= √(0)² + (-6)²

= ** **√0 + 36

= ** **√36

= 6 units

Length of diagonal BD :

Distance between the points B and D

= **√(x₂ - x₁)² + (y₂ - y₁)²**

Here **x₁ = 5, y₁ = 1, x₂ = -3 and y₂ = 1**

**= ** √(-3-5)² + (1-1)²

= √(-8)² + (0)²

= ** **√64

= 8 units

Area of rhombus = (1/2) x 6 x 8

= (1/2) x 48

= 24 square units.

After having gone through the stuff given above, we hope that the students would have understood "How to find the area of a rhombus with vertices".

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