Area Of Quadrilateral

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 (m2). For the computation of area, there are pre-defined formulas for squares, rectangles, circle, triangles, general quadrilaterals etc. In this article, we will learn about the area of a quadrilateral.

Also, read:

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 360o.

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.

Quadrilateral PQRS

  • We can view the quadrilateral as a combination of 2 triangles, with the diagonal PR being the common base.
  • h1 and h2 are the heights of triangles PSR and PQR respectively.

Area Of Quadrilateral

  • 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 * h1)/2
  • Area of triangle PQR = (base * height)/2 = (PR* h2)/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.

Area Of Quadrilateral

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 cm2

Learn more about Quadrilateral, Squares, Rectangle, Rhombus etc. with BYJU’S- The learning app.

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