A quadrilateral is a four-sided polygon, having the sum of interior angles equal to 360^{o}. Quadrilaterals can be classified into different types some of them are square, rectangle, parallelogram, rhombus and trapezoid. Five different formulas are used to calculate the area of the quadrilateral. Let us now discuss the quadrilateral formula in detail.

Source: en.wikipedia.org

**Quadrilateral Formula **

**Perimeter of Quadrilateral**

The perimeter of a quadrilateral is the sum of the length of all sides of the quadrilateral.

The perimeter of the quadrilateral = sum of all sides of a quadrilateral

Suppose ABCD is a quadrilateral,

Perimeter of ABCD = (AB+ BC + CD + DA) units.

**Area of a Quadrilateral**

The Area of a quadrilateral is the number of square units it takes to cover a two-dimensional shape.

Area of a general Quadrilateral = \(\frac{1}{2}×diagonal×(Sum of the height of two triangles)\)

Area of quadrilateral ABCD = \(\frac{1}{2} (h_1 +h_2)\)

where,

d | Diagonal of the quadrilateral |

h_{1} and h_{2} |
heights of the quadrilateral |

**Properties of Quadrilateral**

- Every quadrilateral is a polygon, which has 4 vertices and 4 sides enclosing 4 angles.
- The sum of interior angles of a quadrilateral is 360 degrees i.e. \(\angle1+ \angle2+ \angle3+ \angle4 = 360\).
- In general, a quadrilateral has sides of different lengths and angles of different measures.
- Quadrilateral are of different types that have different properties such as, square, rectangle, parallelogram, rhombus, and trapezoid are special types of quadrilaterals.

### Derivation of Area of Quadrilateral Formula

Consider a quadrilateral ABCD, 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 AC being the common base.
- h
_{1 }and h_{2}are the heights of triangles ADC and ABC respectively. - The area of quadrilateral PQRS is equal to the sum of the area of triangle PSR and the area of triangle PQR.
- Area of triangle ADC = \(\frac{(base × height)}{2}\) = \(\frac{(AC × h_1)}{2}\)
- Area of triangle ABC =\( \frac{(base × height)}{2}\) = \(\frac{(AC× h_2)}{2}\)
- Thus, area of quadrilateral PQRS is,
- Area of triangle PSR + Area of triangle PQR \(\frac{(AC × h_1)}{2}\)+ \(\frac{(AC× h_2)}{2} \)

=\(\frac{ AC(h1+h2)}{2} \)

=\(\frac{1}{2}AC×(h1+h2) \)

Hence, the area of a quadrilateral formula is,

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

**Solved Examples**

**Question 1: **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, h1=5 cm & h2=7 cm

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

Thus, area of quadrilateral ABCD =

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

= \(\frac{(15 × 12)}{2}\) = 90 cm^{2}

**Q2.** Find the perimeter of the quadrilateral with sides 2 cm, 7 cm, 9 cm and 10 cm.

**Solution:**

The formula to find the perimeter of the quadrilateral = sum of the length of all the four sides.

the lengths of all the four sides of a quadrilateral are 2 cm, 7 cm, 9 cm, and 10 cm.

Perimeter of quadrilateral = 2 cm + 7 cm + 9 cm + 10 cm

= 28 cm

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