**Types of Quadrilaterals and Their Properties:** *Different types of quadrilaterals* are explained below with their properties. *Types of polygons* and Types of Quadrilaterals with *Properties of kite Quadrilateral* are also explained.

## Quadrilateral Definition

A quadrilateral is a closed polygon bounded by four straight lines.

A quadrilateral consists of four sides, four edges, and four vertices or corners.

Here, ABCD is a quadrilateral.

AD and BC are diagonals.

∠A+∠B+∠C+∠D =360°

**Different Types of Quadrilaterals and Their Properties**

**Square**

The quadrilateral with all-sided equal and every angle equal to the right angle is a square. Diagonals of a square are equal in length and cut each other at 90°.

**Square sides are equal
**AB=BC=CD=AD

**Square diagonals are equal
**BD=AC = √2 × SIDE

AC and BD bisect each other at O.

**Square Angles**

Angles of a square are equal and 90°. (Each angle)

∠A=∠B=∠C=∠D =90°

**Note
**Square is a special kind of rhombus and rectangles. All the properties of the rhombus and rectangle will be satisfied for the square.

**Rectangle
**The quadrilateral with opposite sides length equal and every angle equal to 90˚ is a rectangle.

Here ABCD is a rectangle.

**Rectangle Sides
**AB = CD and AB || CD

BC = AD and BC || AD

**Rectangle Angles
**∠A = ∠B = ∠C = ∠D = 90° (Each angle)

**Rectangle Diagonals**

AC

^{2 }= AB

^{2 }+ BC

^{2 }= BD

^{2 }= BC

^{2 }+ CD

^{2 }Diagonals AC and BD bisect each other at 90°.

**Rhombus
**Rhombus is a special kind of square with all sides equal, opposite sides parallel, and diagonals bisect each other at 90°.

ABCD is a rhombus.

**Rhombus Sides
**All sides are equal.

AB = BC = CD = AD; AB || CD and BC || AD.

**Rhombus Angles
**Opposite angles are equal.

∠A=∠C and ∠B=∠D

Adjacent angles sum is equal to 180°

∠A + ∠B = 180° = ∠B + ∠C = ∠C + ∠D = ∠D + ∠A

**Rhombus Diagonals
**Diagonals bisect each other at 90°

AD = OC and BO = OD

∠AOB = ∠BOC = ∠COD = ∠DOA = 90°

**Parallelogram
**A parallelogram is a special kind of rectangle with opposite sides parallel and equal in length, opposite angles are equal, adjacent angles sum equal to 180°, and diagonals bisect each other.

**Parallelogram Sides
**Opposite sides are equal and parallel

AB = CD and AB || CD

AD = BC and AD || BC

**Parallelogram Angles
**Adjacent angles sum equals to 180°

∠A + ∠D = ∠D + ∠C = ∠C + ∠B = ∠B + ∠A = 180°

Opposite angles are equal.

∠A = ∠C; ∠D = ∠B.

**Parallelogram Diagonals
**Diagonals bisect each other at “O”

OA = OC and OB = OD but AC || BD.

**Trapezium
**The quadrilateral in which a pair of opposite sides are parallel to each other but they are not equal, that is a trapezium.

ABCD is a trapezium.

AB ||CD but

**AB≠CD**.

**Trapezium Sides
**Two sides are parallel and the other two sides are not parallel AB || OC but

**AB≠DC**.

All the sides lengths are unequal.

**Trapezium Angles
**The Sum of all the angles equals 380°.

**Trapezium Diagonals
**Diagonals bisect each other.

OD = OB and AO = OC.

**Kite Quadrilateral
**A kite is a quadrilateral with two pair of equal length sides which are adjacent to each other.

ABCD is a kite.

**Properties of Kite Quadrilateral**

**Kite Quadrilateral ****Sides
**Two pairs of sides have equal sides and these sides are adjacent to each other.

AB = AD is one pair of sides and AB, AD adjacent to each other.

BC = CD is another pair of sides and BC, CD are adjacent to each other.

**Kite Quadrilateral Angles
**Angles formed by these two pairs of sides are equal.

∠B = ∠D

**Kite Quadrilateral Diagonals
**Diagonals bisect each other at right angles.

OA = OC

OB = OD

**Properties of Quadrilateral**

**Polygon**

Polygon is a 2-dimensional shape bounded by three or more finite number of straight lines. Depending on the number of polygons. For a regular polygon, all the sides are equal.

**Types of Polygons **

POLYGON/SHAPE |
SIDE |

Quadrilateral | 4 |

Pentagon | 5 |

Hexagon | 6 |

Heptagon | 7 |

Octagon | 8 |

Nonagon | 9 |

Decagon | 10 |

These are two types of polygons. They are regular polygons and irregular polygons.

**Regular Polygon
**For a regular polygon, all the side’s length is equal and all the angles are also equal.

Example of regular polygon and pentagon.

The regular pentagon is a regular polygon.

**Irregular Polygon
**For an irregular polygon, all the sides have different lengths and all the angle measurements are unequal.

Example of an irregular polygon.

Ex. Irregular pentagon

**Angles in a Polygon**

**Exterior Angle
**An exterior angle is an angle formed by any side of the polygon and a line extended from the next side.

ABCDEF is a hexagon.

- The Sum of all the exterior angles of a polygon is equal to 360°.
- Each exterior angle of a polygon = 360° / number of sides of the polygon

(360°/6) = 60°

Each exterior angle of the hexagon is 60°.

**Interior Angles
**The Interior angle is an angle formed by two sides of a polygon that share one common point as vertices.

The Sum of an interior angle and exterior angle is equal to 180°.

So, interior angle = 180° – exterior angle.

ABCDEF is a regular hexagon of sides ‘6’.

Interior angles are A, B, C, D, E, and F.

Interior angle = 180° – (360°/n).

Here ‘n’ is ‘6’.

Interior angle = 180° – (360/6) °

= 180° – 60°

= 120°

Each interior angle of the hexagon is 120°.

**Properties of Regular Polygons**

***** ‘n’ is number of sides of the polygon

- Each exterior angle of polygon = (360°/n) or [180° – (Each interior angle)]
- Each exterior angle of a polygon = [(n-2)×180°]/n (or) [180° – (each exterior angle
(or) [180° – (360°/n)]__)]__

- Sum of interior angles of a polygon is (n-2) × 180°
- The Sum of exterior angles of a polygon is (n-2) × 180° / n.
- Number of diagonals of a polygon is [n×(n-3)] / 2.
- Each angle at the center formed by the intersection of lines from any side of a polygon is 360°/n.

θ=360°/n.

**Area of Polygon
**Let polygon be of ‘n’ sided.

- Area of a polygon = (na
^{2 }/4 )×cot(/n). - The radius of inner circle of a polygon is (r)

( a/2)× cot (180°/n) - The radius of outer circle of a polygon is (a/2) × cosec (180°/n)

**Circumcircle, In-Circle, Radius, and Apothem of Polygon**

- The outside circle of a polygon is circumcircle. ‘R’ is the circumradius.
- Circum circle is formed by touching all the vertices in the polygon.

**In-Circle of Polygon**

- Inside circle of a polygon is called in-circle. ‘r’ is the radius of the circle.
- In the circle is formed by touching all the sides of the polygon
- The radius of the circle is also called the apothem.

__ __