# Types of Quadrilaterals and Their Properties | Definition, Formulas

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389 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.

A quadrilateral is a closed polygon bounded by four straight lines.
A quadrilateral consists of four sides, four edges, and four vertices or corners. ∠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 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

Rectangle Angles
∠A = ∠B = ∠C = ∠D = 90° (Each angle)
Rectangle Diagonals
AC2 = AB2 + BC2 = BD2 = BC2 + CD2
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

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.

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

Two pairs of sides have equal sides and these sides are adjacent to each other.
BC = CD is another pair of sides and BC, CD are adjacent to each other.

Angles formed by these two pairs of sides are equal.
∠B = ∠D

Diagonals bisect each other at right angles.
OA = OC
OB = OD

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

1. Each exterior angle of polygon = (360°/n) or [180° – (Each interior angle)]
2. Each exterior angle of a polygon = [(n-2)×180°]/n (or) [180° – (each exterior angle)]
(or)    [180° – (360°/n)]
3. Sum of interior angles of a polygon is (n-2) × 180°
4. The Sum of exterior angles of a polygon is (n-2) × 180° / n.
5. Number of diagonals of a polygon is [n×(n-3)] / 2.
6. 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.

1. Area of a polygon = (na2 /4 )×cot(/n).
2. The radius of inner circle of a polygon is (r)
( a/2)× cot (180°/n)
3. The radius of outer circle of a polygon is (a/2) × cosec (180°/n)

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

Circum Circle 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.