Class 8 Maths Chapter 3 Note - Understanding Quadrilaterals

What are Polygons

A polygon is a closed plane figure with three or more-line segments.

Parts of a Polygon

• Side - Each line segment that forms a polygon. Other name: Edge.
• Vertex - Common endpoint of two sides. Plural: Vertices.
• Angle - Figure formed by two intersecting sides of a polygon, which is measured in degrees.
• Diagonal - Line segment that connects any two non-consecutive or non-adjacent vertices. Number of diagonals in a polygon with n number of sides = n(n-3)/2.

Classification of Polygons

Based on Sides:

A polygon can have infinite number of sides. The polygon with three sides is a triangle. As the number of sides (or vertices) increases, the shape and the name of the polygons changes as mentioned below.

Based on Measurement:

Regular Polygon: All sides and interior angles are of equal measurement. Example: Square, Equilateral triangle.
Irregular Polygon: A polygon with unequal sides or angles or both. Example: Rectangle, rhombus.

Based on Diagonals:

Convex Polygons
• All diagonals are present inside the polygon.
• All interior angles are less than 180º.
• All the vertices always point outward.

Concave Polygons

• A polygon with some diagonals present outside the shape.
• At least one angle measures more than 180º.
• The vertices of a concave polygon point inward as well as outward.


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Angles in a Polygon

Interior angle: An angle formed inside the polygon at the point of intersection of two adjacent sides.

Exterior angle: An angle formed outside the polygon, in between a side and the extension of its adjacent side.

The sum of all the interior angles of a triangle is 180°. There is an easy trick which uses this fact and help us find the sum of the interior angles of a polygon. First, divide the polygon into triangles by drawing diagonals from one single vertex. If we divide a polygon with n sides into triangles by drawing diagonals from one single vertex, the number of triangles formed will be (n – 2). The sum of interior angles of a polygon with n sides is .

So, the sum of interior angles of a pentagon is (5 – 2) × 180°
= 3 × 180°
= 540°

Angles in Regular/Irregular Polygons

The sum of the interior angles for an n-sided regular polygon or an n-sided irregular polygon will be of the same value. This is because the formula of sum of interior angles depends only on the number of sides and here, the number of sides in both polygons is the same.
The interior angle of a regular polygon can be calculated by dividing the sum of its interior angles by its number of sides. The interior angle of a regular polygon with n sides is 180° × (n-2)/n.

The sum of exterior angles of a polygon is always 360○. This is true for all the polygons.

The exterior angle of a regular polygon can be calculated by dividing the sum of its exterior angles by its number of sides. The exterior angle of a regular polygon with n sides is 360°/n.

Properties of Different Quadrilaterals

A quadrilateral is a polygon with four sides and four angles. There are different types of quadrilaterals and all of them show different properties as mentioned below.

1. Parallelogram

• Opposite sides parallel
• Opposite sides equal
• Opposite angles equal
• Diagonals bisect each other

2. Rectangle

• Opposite side parallel
• Opposite sides equal
• Opposite angles equal
• Diagonals bisect each other
• Diagonals equal
• Four right angles

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3. Rhombus

• Opposite side parallel
• Opposite sides equal
• Opposite angles equal
• Diagonals bisect each other
• Diagonals are perpendicular
• Four equal sides

4. Square

• Opposite side parallel
• Opposite sides equal
• Opposite angles equal
• Diagonals bisect each other
• Diagonals equal
• Diagonals are perpendicular
• Four right angles
• Four equal sides

5. Trapezium

• 1 pair opposite sides parallel
• 1 pair opposite sides equal

6. Kite

• Diagonals are perpendicular

The properties can be summarized as below:

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