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A quadrilateral is a four-sided two-dimensional geometrical figure whose sum of all the four internal angles is 360^{o}. Also, it has 4 edges (sides) and four vertices (corners). There are two different types of a quadrilateral namely regular quadrilaterals and irregular quadrilaterals. Few renowned examples of quadrilaterals are square, rectangle, trapezium, rhombus, kite, and parallelogram.

The area of a quadrilateral is the number of square units that can be formed into it. Here, we will look at the important area formulas for quadrilaterals and how to find the area of quadrilateral.

Let us learn to derive the area of the quadrilateral general formula. Consider the quadrilateral PQRS given below:

We can observe the following quadrilateral as a combination of two triangles, considering the diagonal PQ as a common base.

h₁ and h₂ are the heights of triangles PQR and PSR respectively.

Area of quadrilateral PQRS can be calculated by adding the area of two triangles i.e. PQR and PSR.

Let's calculate the area of triangle PQR and the area of triangle PSR.

Area of ΔPSR = 1/2 x Base x Height = 1/2 x PR x h_{1}

Area of ΔPQR = 1/2 x Base x Height = 1/2 x PR x h_{2}

Therefore, the area of quadrilateral PQRS is

Area of ΔPSR + Area of ΔPQR = 1/2 x PR x h_{1} + 1/2 x PR x h_{2} = PR(h_{1} + h_{2}/2)

= 1/2 PR (h_{1} + h_{2})

Hence, the formula to find the area of the quadrilateral is derived as:

Area of General Quadrilateral Formula = 1/2 x diagonals length x ( sum of the height of two triangles ).

The formula to find the area of quadrilateral in trigonometric terms is given as:

Area = ½ x ab x Sin θ

Where a, and b are the length of the diagonals of a quadrilateral and is the angle between them.

In the case of orthogonal quadrilaterals ( such as square, kite, and rhombus), the formulas are minimized to

Area = ½ x ab (as θ is 90^{o}).

If ABCD is a quadrilateral with AC as diagonal, then we can split the quadrilateral into two triangles ABC and ACD.

Now, using the area of triangle formula given its vertices, we can determine the area of triangles ABC and ACD

Therefore, the area of quadrilateral formula coordinate geometry is given as:

Area of Quadrilateral ABCD = Area of Triangle ABC + Area of Triangle ACD

Using this information, we can find the area of a quadrilateral if its vertices are given:

Let the vertices of quadrilateral ABCD be A ( x₁,y₁), B(x₂,y₂), C(x₃,y₃), D(x₄,y₄)

Area of Quadrilateral ABCD = Area of Triangle ABD + Area of Triangle BCD

+1/2{(x₁y₂ + x₂y₄ + x₄y₁) - (x₂y₁ + x₄y₂ + x₁y₄)}

= 1/2{(x₂y₃ + x₃y₄ + x₄y₂) - (x₃y₂ + x₄y₃ + x₂y₄)}

= 1/2{(x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) - (x₂y₁ + x₃y₂ + x₄y₃ + x₁y₄)}

=1/2(x₁ - x₃) (y₂ - y₄) -(x₂ -x₄) (y₁ - y₃) sq.units

Here is the list of the formulas to find the area of quadrilaterals such as square, kite, parallelogram, trapezium, rectangle, and rhombus.

Here, we have discussed the area formula for different types of quadrilaterals. These quadrilateral formulas help you to calculate the area of square, rectangle, parallelogram, kite, rhombus, and trapezium. We have also discussed the area of quadrilateral formula coordinate geometry that is derived by dividing the quadrilateral into two triangles, calculating the area of each triangle given its vertices, and adding these values to get the total area of the quadrilateral.

FAQ (Frequently Asked Questions)

1. Define Quadrilateral.

Ans. In Geometry, a quadrilateral is a four-sided two- dimensional shape that is formed by joining four distinct points among which any three points are noncollinear.

2. Define the Area of the Quadrilateral.

Ans. The area of the quadrilateral is defined as the space occupied by the quadrilateral shape in two-dimensional space.

3. How to Find the Area of a Quadrilateral When the Length of the Diagonal and Length of the Perpendicular From the Opposite Vertices are Given?

Ans. If the length of the diagonal and length of the perpendicular from the opposite vertices are given, then the area of the quadrilateral can be calculated as:

Area of Quadrilateral = 1/2 x length of the diagonals x ( sum of the length of the perpendicular drawn from the remaining two vertices.

4. What are the Different Types of Quadrilaterals?

Ans. The different types of quadrilaterals are rhombus, square, rectangle, kite, trapezium, and parallelogram.