Quadrilateral Formulas

A quadrilateral is a closed figure and a type of polygon which has four sides or edges, four angles, and four corners or vertices. The word quadrilateral is derived from the Latin words “quad”, a variant of four, and “latus”, meaning side. It is also called a tetragon, derived from Greek word “tetra”, meaning four, and “gon” meaning corner or angle.  

A quadrilateral is formed by joining four non-collinear points. The sum of interior angles of a quadrilateral is always equal to 360 degrees.

It is not necessary that all four sides of a quadrilateral are equal in length. Thus, we can have different types of quadrilaterals based on their sides and angles.

Here, ABCD is a quadrilateral, with four sides namely AB, BC, CD, DA, four angles ∠A, ∠B, ∠C, ∠D, and the lines joining A to C and B to D are two diagonals of the quadrilateral.

Types of Quadrilateral 

1. Parallelogram
2. Rectangle
3. Square
4. Rhombus
5. Trapezium
6. Kite

Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are of equal length and the opposite angles of a parallelogram are also equal.

Rectangle 

A Rectangle is a type of quadrilateral that has its parallel sides equal to each other and all four angles 90°. Hence, it is also called an equiangular quadrilateral.

Square

A quadrilateral with four equal sides and four equal angles is called a Square.

Rhombus

A Rhombus is a type of parallelogram with four equal sides and equal opposite angles.

Trapezium

A trapezium is a kind of quadrilateral having only one pair of sides parallel to each other. 

Kite

A quadrilateral having pair of adjacent sides equal is known as Kite.

Formulas of Quadrilateral

There are two basic formulas for quadrilaterals

1. Area
2. Perimeter

Area of Quadrilaterals

In geometry, the area can be defined as the space occupied by a flat shape or the surface of an object.  The area of a figure is the number of unit squares that cover the surface of a closed figure. The area is measured in square units such as square centimeters, square feet, square inches, etc. 

Area of Parallelogram = Base × Height

Area of Rectangle = Length × Width

Area of Square = Side × Side

Area of Rhombus = 1/2 × diagonal1 × diagonal2

Area of Trapezium = 1/2 × Height × (Length1 + Length2)

Area of Kite = 1/2 ×diagonal1 × diagonal2

Perimeter of Quadrilaterals

In geometry, the perimeter can be defined as the path or the boundary that surrounds a shape. It can also be defined as the length of the outline of a shape.

Since we know that quadrilateral has four sides, therefore, the perimeter of any quadrilateral say, ABCD, is given by

Perimeter = AB + BC + CD + DA (sum of all the sides)

Perimeter of Parallelogram = 2×(Base + Side)

Perimeter of Rectangle = 2×(Length + Width)

Perimeter of Square = 4 × Side

Perimeter of rhombus = 4 × Side

Perimeter of Trapezium = Sum of all the sides

Perimeter of Kite = 2×(a + b), where a, and b are adjacent pairs

Sample Problems

Problem 1: If 20cm and 10cm are diagonal lengths of a kite, then find the area of the kite.

Solution:

Given : 

Length of diagonal1 = 20cm

Length of diagonal2 = 10cm

Area of Kite =1/2 × diagonal1 × diagonal2

Area =1/2 ×20 ×10 = 100cm²

Problem 2: How can we find the perimeter of an irregular Quadrilateral?

Solution:

To determine the perimeter of an irregular quadrilateral we can simply add the length of the outer sides of the quadrilateral. Because perimeter is nothing but the total length of the periphery of any shape.

Problem 3: Find the area of the trapezium whose length of parallel sides is 7cm and 18cm respectively and the height of the trapezium is 10cm.

Solution:

Given,

Length of parallel sides of Trapezium,

Length 1 = 7cm

Length 2 = 18cm

Height of Trapezium = 10cm

we know that, Area of Trapezium = 1/2 × Height × (Length1 + Length2)

Therefore,

Area = 1/2 × 10 ×(7 +18)

          =125cm²

Hence, Area of the given trapezium is 125cm²

Problem 4: The perimeter of a quadrilateral is 90cm and the length of the three sides are AD = 23cm, AB = 28cm and BC = 18cm. Find the length of the fourth side i.e, CD.

Solution:

Given,

Length of side AB = 28cm

Length of side BC = 18cm

Length of side AD = 23cm

Let the length of side CD = x cm

we know that,

Perimeter = AB + BC + CD + AD

This implies,

90 = 28 + 18 + x +23

90 = 69 + x

x = 21

Hence, the length of side AD = 21cm

Problem 5: If the area of a rhombus is 70cm² and the base is 15cm, then find out the height of the given rhombus.

Solution:

Area = 70cm²

Base = 15cm

Since Area of Rhombus = Height × Base

This implies, 

70 = Height × 15

Height = 70/15

Height = 4.67cm

Problem 6: Write down the formula to calculate the length of the diagonal of a rectangle.

Solution:

The diagonal of a rectangle is a line segment drawn to connect any two non-adjacent vertices of a rectangle. A rectangle can have a maximum of two diagonals of equal length.

A diagonal rectangle divides the rectangle into two right-angle triangles. Therefore we can easily calculate the length of diagonals using the Pythagoras Theorem, where the diagonals are considered as the hypotenuse of the right triangle.

Consider triangle BCD,

Since the triangle BCD is a right angle triangle,

Therefore, (BC)² = (BD)² + (CD)²

                 (BC)² = (width)² + (length)²

                  BC = √(width)² + (length)²

Problem 7: Find the perimeter of a parallelogram whose base is 12cm and height is 23cm.

Solution:

Base length of given parallelogram = 12cm

Height of given parallelogram = 23cm

Perimeter of a parallelogram = 2×(a + b)

where a = 12cm and b = 23cm

Perimeter of parallelogram = 2×(12 + 23)

                                            = 70cm

Hence, the perimeter of the given parallelogram is 70cm