GMAT / GRE Mathematics Tutorial Series: Quadrilaterals

This tutorial is based on geometry concepts. We will study the types and properties of five quadrilaterals: Parallelogram, rectangle, square, rhombus and trapezium.

Basic definition

In Euclidean plane geometry, a quadrilateral is basically a four sided polygon, with the internal angles adding up to 360°.

Quadrilateral

Quadrilateral

Types of Quadrilateral

GMAT Exam Quadrilaterals tutorials are of five types based on their shapes. These are:

  • Parallelogram
  • Rectangle
  • Square
  • Rhombus
  • Trapezium

quadrilateral types

We’ll study each of these quadrilaterals in detail.

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Parallelogram

A parallelogram is a quadrilateral with opposite sides parallel (as seen in the figure below).

Parallelogram

Properties of a parallelogram

  • Opposite angles are equal.
  • Opposite sides/facing sides are parallel and of equal length.
  • Adjacent angles are supplementary, i.e. they add up to 180°.
  • Each diagonal divides the parallelogram into two equal triangles. The two diagonals bisect each other.
  • If one of the angles of a parallelogram is a right angle then the parallelogram is a rectangle.

Important formulas

If the length of a parallelogram is ‘l’, breadth is ‘b’ and height is ‘h’ then:

  • Area = L * H
  • Perimeter = 2(L+B)

Parallelogram

Rectangles

A rectangle is a quadrilateral with all angles at right angles (360°/4 = 90°). (As seen in the figure below)

Rectangle

Properties of a Rectangle

  • Opposite sides are both parallel and equal to each other.
  • All angles are 90°.
  • The diagonals are equal and bisect each other (divide each other equally).
  • Opposite angles formed at the point where diagonals meet are equal.
  • A rectangle is a parallelogram with all the angles 90°.

Important formulas

  • If L = length and B = breadth, then

Length of the diagonal = √ (L2 + B2)

  • Area = L * B
  • Perimeter = 2(L+B)

Rectangle

Squares

SquareA square is a quadrilateral in which all the sides are equal (as seen in the figure above).

Properties of a square

  • All sides and angles are equal.
  • Opposite sides are parallel to each other.
  • The diagonals are equal and bisect each other at right angles.
  • A square is a type of parallelogram in which all the angles and sides are equal.
  • Also, a parallelogram becomes a square when the diagonals are equal and right bisectors of each other.

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Important formulas for Squares

  • If ‘L’ is the length of the side of a square then length of the diagonal = L √2.
  • Area = L2.
  • Perimeter = 4L

Square

Rhombus

Rhombus

A rhombus is a quadrilateral having equal sides and opposite sides are parallel to each other. (As seen in the figure above)

Properties of a Rhombus

  • All sides are equal
  • Opposite angles are equal
  • The diagonals bisect each other at right angles
  • Adjacent angles are supplementary (e.g. ∠A + ∠B = 180°)
  • A rhombus is also a parallelogram that has diagonals that are perpendicular to each other
  • A rhombus with right angles is a square

Important formulas for a Rhombus

If a and b are the lengths of the diagonals of a rhombus,

  • Area = (a* b) / 2
  • Perimeter = 4L

Rhombus

Trapezium

A trapezium or a trapezoid is a quadrilateral that has only one pair of parallel sides called bases and two lateral sides called legs.

Trapezium

Properties of a Trapezium

  • Only the bases of the trapezium are parallel to each other (MN ⫽ OP).
  • None of the sides, angles and diagonals are equal.

Important Formulas for a Trapezium

  • Area = (1/2) h (L+L2)
  • Perimeter = L + L1 + L2 + L3 

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Summary of properties

Summarizing what we have learned so far for quick reference and recollection:

 

S.No.PropertyParallelogramRectangleRhombusSquare
1All sides are congruent✕✕✓✓
2Opposite sides are parallel and congruent✓✓✓✓
3All angles are congruent✕✓✕✓
4Opposite angles are congruent✓✓✓✓
5Diagonals are congruent✕✓✕✓
6Diagonals are perpendicular✕✕✓✓
7Diagonals bisect each other✓✓✓✓
8Adjacent angles are supplementary✓✓✓✓

 

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