Quadrilateral

In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word ‘quadrilateral’ is derived from the Latin words ‘quadri,’ which means four, and ‘latus’, which means side.

Parts of a quadrilateral

  • ∠A, ∠B, ∠C, and ∠D are the four angles of the quadrilateral ABCD
  • AB, BC, CD, and DA are the four sides of the quadrilateral ABCD.
  • A, B, C, and D are the four vertices of the quadrilateral ABCD.
  • AC and BD are the two diagonals of the quadrilateral ABCD.

Properties of Quadrilaterals

Some properties are common to all quadrilaterals. These properties are:

  • They have four vertices.
  • They have four sides.
  • The sum of all interior angles is 360°.
  • They have two diagonals.
  • A quadrilateral can be regular or irregular. A regular quadrilateral must have 4 equal sides, and 4 equal angles, and its diagonals must bisect each other. Square is the only quadrilateral that satisfies all these conditions.

Types of Quadrilaterals

Quadrilaterals can be classified into Parallelograms, Squares, Rectangles, and Rhombuses. Square, Rectangle, and Rhombus are also Parallelograms.

  • Parallelogram: a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of equal length; that opposite angles are equal; or that the diagonals bisect each other. Parallelograms include rhombi (including those rectangles called squares) and rhomboids (including those rectangles called oblongs). In other words, parallelograms include all rhombi and all rhomboids, and thus also include all rectangles.
  • Rhombus: all four sides are of equal length (equilateral). An equivalent condition is that the diagonals perpendicularly bisect each other.
  • Rhomboid: a parallelogram in which adjacent sides are of unequal lengths, and some angles are oblique (equiv., having no right angles). 
  • Rectangle: all four angles are right angles (equiangular). An equivalent condition is that the diagonals bisect each other, and are equal in length. 
  • Square (regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are right angles. An equivalent condition is that opposite sides are parallel (a square is a parallelogram), and that the diagonals perpendicularly bisect each other and are of equal length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle (i.e., four equal sides and four equal angles).
  • Kite: two pairs of adjacent sides are of equal length. This implies that one diagonal divides the kite into congruent triangles, and so the angles between the two pairs of equal sides are equal in measure. It also implies that the diagonals are perpendicular. Kites include rhombi.

Concave and Convex Quadrilaterals

Concave quadrilaterals: In concave quadrilaterals, one interior angle is greater than 180°. 

A quadrilateral is called a concave quadrilateral if at least one diagonal, i.e. the line segment joining the vertices is not a part of the same region of the quadrilateral.

Convex quadrilaterals:  In convex quadrilaterals, each interior angle is less than 180°. A quadrilateral is convex if the line segment joining any of its two vertices is in the same region.

Perimeter of Quadrilateral

The perimeter of a quadrilateral is the length of its boundary. This means the perimeter of a quadrilateral equals the sum of all the sides. If ABCD is a quadrilateral then its perimeter will be: AB + BC + CD + DA

erimeter of quadrilateral ABCD = AB + BC + CD + DA

The formula for the perimeter of some of the common quadrilaterals is given below:

Rectangle 2 (length + width)
Square 4 x Side
Rhombus 4 x Side
Parallelogram 2 x sum of adjacent sides
Kite 2 x sum of adjacent sides

Area of Quadrilateral

The area of the quadrilateral is the region enclosed by all its sides. The formulas to find out the area of different types of quadrilaterals are shown below:

  • If the length of the rectangle is L and breadth is B then, Area of a rectangle = Length × Breadth or L × B
  • If the side of a square is ‘a’ then, Area of the square = a × a = a²
  • If the length of a parallelogram is ‘l’, breadth is ‘b’ and height is ‘h’ then: Area of the parallelogram = l × h
  • If the length of two diagonals of the rhombus is d1 and d2 then the area of a rhombus = ½ × d1 × d2

If the height of a trapezium is ‘h’ (as shown in the diagram) then: Area of the trapezium = ½ × (Sum of lengths of parallel sides) × h = ½ × (AB + CD) × h

  • The area of a kite is half the product of the lengths of its diagonals. The formula to determine the area of a kite is: Area = ½ × (d)1 × (d)2. Here (d)1 and (d)2 are long and short diagonals of a kite.