Quadrilaterals

In geometry, a quadrilateral is a closed shape that is formed by joining four points among which any three points are non-collinear. A quadrilateral has 4 sides, 4 angles, and 4 vertices. 'Quadrilateral' is derived from a Latin word, in which, 'Quadra' means four and 'Latus' means sides. All 4 sides of a quadrilateral may or may not be equal.

A quadrilateral is a polygon with four sides four angles and four vertices. Whenever we name a quadrilateral, we need to keep in mind the order of the vertices. For example, the following quadrilateral should be named as ABCD, BCDA, ADCB, or, DCBA. It cannot be named as ACBD or DBAC, since they change the order of vertices in which a quadrilateral is formed. The following quadrilateral ABCD has four sides: AB, BC, CD, DA and two diagonals: AC and BD.

Although a quadrilateral always has four sides, four angles, and four vertices, the measure of the sides and angles differ. It is to be noted that the sum of the interior angles of a quadrilateral is always equal to 360°. The following table lists the different types of quadrilaterals. 

Each of the quadrilateral discussed above has its own properties. Though, there are some properties that are common to all quadrilaterals. They are as follows.

• They have four sides.
• They have four vertices.
• They have two diagonals.
• The sum of all interior angles is 360°.

We will study the other properties of different quadrilaterals in detail. We can identify a quadrilateral by using the following properties of quadrilaterals.

Square

A square is a quadrilateral with four equal sides and four right angles.

A square has:

• Four equal sides
  AB = BC = CD = DA
• Four right angles
  ∠ A= ∠B = ∠C = ∠D = 90°
• Two pairs of parallel sides
  AB ∥ DC and AD ∥ BC
• Two equal diagonals
  AC = BD
• Diagonals that are perpendicular to each other
  AC ⊥ BD
• Diagonals that bisect each other.

Rectangle

A rectangle is a quadrilateral with two pairs of equal and parallel opposite sides and four right angles.

A rectangle has:

• Two pairs of parallel sides
  AB ∥ DC and AD ∥ BC
• Four right angles
  ∠A = ∠B = ∠C = ∠D = 90°
• Opposite sides of equal lengths
  AB = DC and AD = BC
• Two equal diagonals
  AC=BD
• Diagonals that bisect each other.

Parallelogram

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. 

A parallelogram has:

• Two pairs of parallel sides
  PQ ∥ RT and PR ∥ QT
• Opposite sides of equal lengths
  PQ = RT and PR = QT
• Opposite angles that are equal
  ∠P  = ∠T and ∠Q = ∠R
• Two diagonals that bisect each other.

Trapezium

A trapezium is a quadrilateral in which one pair of opposite sides is parallel. 

• The sides that are parallel to each other are called bases.
  In the above figure, EF and GH are the bases.
   
• The sides that are not parallel to each other are called legs.
  In the above figure, EG and FH are legs.

There is nothing special about the sides, angles, or diagonals of a trapezium. But if the two non-parallel opposite sides are of equal length, then it is called an isosceles trapezium. The following quadrilateral XYZW is an isosceles trapezium, in which the legs are equal, i.e., WX = ZY, and the diagonals are also equal, i.e., XZ = WY.

Rhombus

A rhombus is a quadrilateral with four equal sides.

A rhombus has:

• Two pairs of parallel sides
  EH ∥ FG and EF ∥ HG
• Four equal sides
  EH = HG = GF = FE
• Opposite angles that are equal
  ∠E = ∠G and ∠H = ∠F
• Diagonals that are perpendicular to each other
  EG ⊥ HF
• Diagonals that bisect each other.

Kite

A kite is a quadrilateral in which two pairs of adjacent sides are equal. 

Observe the following properties of a kite. A kite has:

• Two pairs of equal adjacent sides
  AB = BC and CD = DA
• One pair of opposite angles (which are obtuse) that are equal
  ∠A = ∠C
• Diagonals that are perpendicular to each other
  AC ⊥ BD
• A longer diagonal that bisects the shorter diagonal.

Think Tank

• Can a kite be called a parallelogram?
• What elements of a trapezium should be changed to make it a parallelogram?

Area of Quadrilaterals

The area of a quadrilateral is the number of unit squares that can be fit into it. The following table lists the formulas for finding the area of quadrilaterals.

Topics Related to Quadrilaterals

Check out some interesting articles related to quadrilaterals.

• Geometric Shapes
• Triangles
• Euclidean Geometry
• Solid Shapes
• Angles
• Parallelograms
• Angle Sum Property in Quadrilaterals 

What is a Quadrilateral?

A quadrilateral is a closed two-dimensional figure that has 4 sides, 4 angles, and 4 vertices. A few examples of quadrilaterals are square, rectangle and trapezium.

What are the Different Types of Quadrilaterals?

There are different types of quadrilaterals that are identified on the basis of their unique properties. For example, square, rectangle, parallelogram, rhombus, kite, trapezium, isosceles trapezium are all categorized under quadrilaterals.

What is the Sum of the Interior Angles in a Quadrilateral?

In any type of quadrilateral, the sum of the interior angles is always equal to 360°. For example, a rectangle is a quadrilateral with each of its interior angles equal to 90° which makes it (90 × 4) = 360°.

What are All the Properties that Quadrilaterals Share in Common?

Though there are different types of quadrilaterals, they share a few properties that are common. They are listed as follows:

• They have four sides.
• They have four vertices.
• They have two diagonals.
• The sum of all interior angles is 360°.

How to Find the Area of a Quadrilateral?

Area is the space that is covered by any shape. A quadrilateral is a figure that has 4 sides and the space occupied by a quadrilateral is called its area. A few examples of quadrilaterals are square and rectangle. The area of a square of side 'a' is calculated by the formula: Area = 'a × a' or a², and the area of a rectangle whose length is 'l' and width is 'w' is calculated by the formula: Area = 'l × b'. The area of a quadrilateral is always expressed in square units.

How do you Find the Perimeter of a Quadrilateral?

Perimeter is the length of the boundary of any shape. As we know that a quadrilateral has 4 sides, the perimeter of a quadrilateral can be found by adding all the sides of the quadrilateral. For example, if a rectangle has a length of 6 units and a width of 4 units then we use the formula for the perimeter of a rectangle which is:  2(length + width). Substituting the values in the formula:  2 (6 + 4) = 20 units.