GRE Geometry | Quadrilaterals

A quadrilateral is a 2 - D shape which contain 4 sides. It consist of four sides, four angles and four vertices. 
 

Different type of quadrilateral and their properties: 

• Rectangle: 
  The rectangle, like the square, is one of the most commonly known quadrilaterals. It is defined as having all four interior angles 90° (right angles). 
  
  
   

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  • All four angles in a rectangle are 90°. 
     
  • Opposite side are parallel and of same length(or congruent). 
     
  • Diagonals of rectangle bisect each other and divide a the rectangle into two congruent triangles. 
     
  • Area of rectangle = length * breadth. 
     
  • Perimeter of rectangle = 2*(length + breadth)
• Square: 
  In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. 
  
   

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  • All four angles in a rectangle are 90°. 
     
  • All sides are equal and opposite sides are parallel(or congruent). 
     
  • Diagonals of square bisect each other at 90° and divide the square into two congruent triangle. 
     
  • Area of square = side * side 
     
  • Perimeter of square = 4 * side
• Rhombus: 
  A Rhombus is a flat shape with 4 equal straight sides. A rhombus looks like a diamond. All sides have equal length. Opposite sides are parallel, and opposite angles are equal (it is a Parallelogram). 
  
   

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  • All sides are equal in a rhombus. 
     
  • Opposite angles are equal in rhombus. 
     
  • The diagonals of rhombus intersect each other at equal angles. 
     
  • Area of rhombus = (diagonal₁ * diagonal₂) / 2 
     
  • Perimeter of rhombus = 4 * side
• Trapezium: 
  In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezoid. 
  
   

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  • Atleast one pair of opposite side is parallel to each other. 
     
  • Area of trapezium = (sum of length of parallel side(or bases)) * height / 2. 
     
  • Perimeter of trapezium = sum of all sides. 
     
• Parallelogram: 
  In Euclidean geometry, a parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. 
  
   

• 
  
  • Opposite sides are parallel and congruent to each other. 
     
  • Opposite angles are congruent to each other. 
     
  • Sum of adjacent angles is 180° 
     
  • Diagonals of parallelogram bisect each other and divide the parallelogram into two congruent triangles. 
     
  • Area of parallelogram = length * breadth 
     
  • Perimeter of parallelogram = 2(length + breadth) 
     

Some examples on quadrilateral: 
What will be the area and perimeter of given rectangle if length of rectangle is 5cm and breadth of rectangle is 7cm? 
 

Solution: 
 

Area of rectangle = length * breadth. 
Area of rectangle = 5 * 7 = 35 cm2
Perimeter of rectangle = 2*(length + breadth)
Perimeter of rectangle = 2*(5 + 7) = 24cm

What will be the area of given rhombus if diagonal of rhombus are 4cm and 6cm? 
 

Area of rhombus = (diagonal1 * diagonal2) / 2
Area of rhombus = (4 * 6) / 2 = 12 cm2

What will be area and perimeter of given trapezium if AB is parallel to CD? 

Area of trapezium = (sum of length of parallel side(or bases)) * height / 2.
Area of trapezium = (3 + 4. 5) * 3 / 2 
= 11.25 cm2
Perimeter of trapezium = Sum of all sides
= 3 + 3 + 3.5 + 4.5 = 14 cm