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 quadrilateral.

What is a Quadrilateral?

[Before going into the calculation of area, let usĀ  define what is quadrilateral. A quadrilateral is a four-sided polygon, having the sum of interior angles equal to \(360^{\circ}\).

Properties of a 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

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

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

Let us solve few examples-

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

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Practise This Question

Find the area of the given quadrilateral ABCD. Given that the length of BD, AE and BC are 28 cm, 12 cm, and 8 cm respectively.