You are already acquainted with the term area. It is defined as the region occupied inside the boundary of a flat object or figure. The measurement is done in square units with the standard unit being square metres (m^{2}). For the computation of area, there are pre-defined formulas for squares, rectangles, circles, triangles, general quadrilaterals etc. In this article, we will learn about the area of a quadrilateral.

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## What is a Quadrilateral?

Before going into the calculation of area, let us define what is a quadrilateral. A quadrilateral is a four-sided polygon, having the sum of interior angles equal to 360^{o}.

## Properties of Quadrilateral

- Every quadrilateral has 4 vertices and 4 sides enclosing 4 angles.
- The sum of its interior angles is 360 degrees.
- A quadrilateral, in general, has sides of different lengths and angles of different measures. However, squares, rectangles, parallelograms, etc. are special types of quadrilaterals with some of their sides and angles being equal.

## Area of Quadrilateral Formula

Consider a quadrilateral PQRS, of different (unequal) lengths, let us derive a formula for the area of a quadrilateral.

- We can view the quadrilateral as a combination of 2 triangles, with the diagonal PR being the common base.
- h
_{1 }and h_{2}are the heights of triangles PSR and PQR, respectively.

- Area of quadrilateral PQRS is equal to the sum of the area of triangle PSR and the area of triangle PQR.
- Area of triangle PSR = (base * height)/2 = (PR * h
_{1})/2 - Area of triangle PQR = (base * height)/2 = (PR* h
_{2})/2 - Thus, area of quadrilateral PQRS is,
- Area of triangle PSR + Area of triangle PQR =\(\frac{PR \times h_{1}}{2} + \frac{PR \times h_{2}}{2} = PR \left ( \frac{h_{1}+ h_{2}}{2} \right )\)
- \(= \frac{1}{2} PR \times (h_{1}+ h_{2})\)

Hence, the area of a quadrilateral formula is,

Area of a general Quadrilateral \(= \frac{1}{2} \times \; diagonal \times (Sum \; of \; height \; of \; two \; triangle)\)

### Area of a Quadrilateral Example

**Question: **

In the given quadrilateral ABCD, the side BD = 15 cm and the heights of the triangles ABD and BCD are 5 cm and 7 cm, respectively. Find the area of the quadrilateral ABCD.

**Solution: **

Diagonal = BD = 15 cm

Heights, \(h_{1} = 5\) cm & \(h_{2} = 7\) cm

Sum of the heights of the triangles = h1 + h2 = 5 + 7 = 12 cm

Thus, area of quadrilateral ABCD =

\(= \frac{1}{2} \times \; diagonal \times (Sum \; of \; height \; of \; two \; triangle)\)

= (15 * 12)/2 = 90 cm^{2}

## Frequently Asked Questions on Area of Quadrilateral

### What is the area of a quadrilateral?

The area of the quadrilateral is the space occupied by the shape quadrilateral in the two-dimensional space. As we know, a quadrilateral is a 2D figure with four sides. Generally, a quadrilateral is the combined form of a regular or an irregular triangle.

### Mention the different types of quadrilateral.

The different types of a quadrilateral are:

Square

Rectangle

Rhombus

Kite

Parallelogram

Trapezium

### How to calculate the area of a quadrilateral?

The quadrilateral is the combination of the basic geometric shape called triangles. To calculate the area of a quadrilateral, the area of the individual triangles should be computed, and add the area of the individual triangles.

### Mention the applications of quadrilaterals.

Quadrilaterals and its area are mostly used in the field of architecture, agriculture, designing, and navigation to find the actual distance with precision.

### How to calculate the area of a quadrilateral if one of its diagonals and perpendiculars from the vertices are given?

If the diagonal and the length of the perpendiculars from the vertices are given, then the area of the quadrilateral is calculated as:

Area of quadrilateral = (Â½) Ã— diagonal length Ã— sum of the length of the perpendiculars drawn from the remaining two vertices.

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