Understanding Quadrilaterals

Any closed polygon having 4 sides, 4 angles and 4 vertices is called the Quadrilateral. A quadrilateral could either be a regular or an irregular polygon. The Angle sum property of a Quadrilateral is dependent on the sum of the exterior and the interior angles. It says that the sum of all the exterior angles of a Quadrilateral equals 360°. Even the sum of all the interior angles of a Quadrilateral is equal to 360°. 

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For understanding quadrilaterals class 8, learn about these geometrical figures for a thorough knowledge.


Understanding Quadrilaterals Class 8 Exercise 3.1

  1. Plane Surface

A surface that’s completely flat like a chequered board or a paper is a plane surface.

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  1. Plane Curve

When we obtain a curve by connecting the number of certain points without picking the pencil up is a plane curve. A plane curve can be classified into 2 i.e. an open or closed curve.

Open Curve

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Closed Curve

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  1. Polygons

The simple closed curves which are solely composed of line segments are called the Polygons.

Polygons are further classified by the number of sides or vertices they have. Take a look at the chart to understand all types of polygons

Number of Sides

Type of Polygon

Geometrical Figure

3

Triangle

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4

Quadrilateral

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5

Pentagon

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6

Hexagon

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7

Heptagon

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8

Octagon

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9

Nonagon

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10

Decagon

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n

n-gon

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Understanding Quadrilaterals Class 8 Exercise 3.2

Classify each of the following figures on the basis of the given:-

(i) Polygon

(ii) Convex polygon

(iii) Concave polygon

(iv) Simple Curve

(v) Simple Closed Curve

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Lets unlock the answers to above;-

(i) 1, 2

(ii) 2

(iii) 1

(iv) 1,2,5,6,7

(v) 1,2,5,6,7


Concave and Convex Polygons

Concave Polygon

The concave polygons are the type of polygons that have some of its diagonals in the exterior of the object.  


Convex Polygon

The convex polygons are the type of polygons that have all of its diagonals in the interior of the object.  


Regular and Irregular Polygons

Polygons that are equilateral as well as equiangular are known as Regular Polygons. In other words, a polygon is regular if-

  • It’s all (four) sides are equal.

  • It’s all (four) angles are equal.

Therefore, we can conclude that a square is a regular polygon but a rectangle is not since all of its sides are not equal, though the angles are equal.


Angle Sum Property of a Polygon

The sum total of all the interior angles of a polygon stays the same as per the number of sides irrespective of the shape of the polygon.

That being said, the sum of interior angles of a polygon is-

[n – 2] × 180°

Where n refers to - number of sides of the polygon

As an example, for quadrilateral [4-2] × 180° = 360°

Note: The angle sum property of a polygon is applicable to both concave and convex polygon.


Solved Examples

Let’s practice some quadrilateral problems and solution of understanding quadrilaterals class 8

Example1:

Find out the sum of the measures of the angles of a convex quadrilateral? Reasons why this property holds true if the quadrilateral is not convex? (Draw a non-convex quadrilateral and try!)

Solution1:

According to the Angle sum of a convex quadrilateral = (4 – 2) × 180° = 2 × 180° = 360°    

Seeing that, a quadrilateral, which is concave, i.e. not convex has a similar number of sides i.e. 4 as a concave quadrilateral have, therefore, a quadrilateral which is not convex also holds this property. I.e. angle sum of a non-convex (concave) quadrilateral also equal to 360°

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Above is a quadrilateral with sides ABCD, made of two triangles ▴ABC and ▴ADC. This means that the sum of all the interior angles of this quadrilateral will be equal to the sum of interior angles of two triangles i.e. 180° + 180° = 360°

Yes, the property holds true for a quadrilateral that is not convex since any quadrilateral can be split up into two triangles.  

Our quadrilateral is also concave, made of two triangles ▴ABC and ▴ADC.

 

Example2:

Find out the angle sum of a polygon i.e. a heptagon with 7 numbers of sides?

Solution2:

Given number of sides = 7      

Applying the rule to Angle sum of a polygon with 7 sides

= (7 – 2) × 180°

= 5 × 180°

= 900°   

FAQ (Frequently Asked Questions)

1. How Many Kinds of Quadrilaterals Are There?

There are a number of quadrilaterals that are named based on the characteristic of their sides and their angle. Check below the different types of quadrilateral;-


A. Trapezium

A trapezium is a type of a quadrilateral with one pair of parallel sides. Note that If the non-parallel sides of a trapezium measure the same then it is called an Isosceles Trapezium.


B. Kite

A kite is a type of a quadrilateral with two pairs of adjacent sides that measure the same.


C. Parallelogram

A parallelogram is a type of a quadrilateral with two pairs of opposite sides that are parallel to each other.


Remember that a rhombus, rectangle, and a square are also quadrilateral which also makes for a special type of parallelograms.  


2. What Does a Measure of Sum of the Exterior Angles of a Polygon Dictate?

  • The total of the exterior angles of any polygon equals 360°.

  • It is applicable to regular and irregular, small and large polygons.

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Sum of all the exterior angles in the given figure of irregular pentagon is as follows;-

59° + 50° + 35° +61° = 360° - y°

Thus,

Y° = 155°

Now, the sum of all exterior angles of a pentagon is 59° + 50° + 35° +61°+ 155°= 360°

Note: This property is used to compute the number of sides in a regular polygon. The property is pertinent to irregular polygons also. The sum will remain the same regardless of it being a regular or irregular, small or big polygon.