3. Dear children.
We use a lot of math, knowingly or unknowingly, in our
daily life. We try to discuss such mathematical ideas from this class onwards.
By looking a little deeper into ideas, this book tries to develop the habit of
generalization.
In it, you will find examples which make you think about
the relations between cause and effect. It also gives you various opportunities
to form your own opinions and express them with confidence.
With the help of the teacher, you can observe things in
new ways, discover new facts through experiments and find the reasons for
many things.
Let the learning of math be full of joy for you.
With love,
DIRECTOR
SCERT
4. 1. POLYGON
INTRODUCTION
POLYGON
It is a closed plane figure, bounded by straight-lie segment.
The line segments forming a polygon intersect only at end
Points and each end point are shared by only two line segments.
The segments which make up a polygon are called the sides of the polygon
and the end points of the segments are called the vertices of the polygon.
The polygons are named according to the number of sides they contain.
e.g.
3 4 5 6
SIDES SIDES SIDES SIDES
TRIANGLE QUADRILATERAL PENTAGON HEXAGON
5. Remember:
A line segment joining any two consecutive vertices of a polygon is called its
diagonal.
In the adjoining figure, AC is a diagonal of pentagon ABCDE as it joins to
non- consecutive vertex A and C of the pentagon. Similarly, AD is a diagonal.
More diagonals can be drawn through the vertices B, C, D and E of the
pentagon ABCDE.
E
D
A
B C
SUM OF ANGLES OF A POLYGON
Draw all possible diagonals through a single vertex of a polygon to form as
many triangles as possible.
4 Sides 5 Sides 6 Sides
2 Triangles 3 triangles 4 triangles
6. It is observed that the number of triangles formed is two less than the number
of sides in the polygon.
So, if a polygon n sides, the number of triangles formed will be n-2.
Since, The sum of the angles of a triangle = 180^0
There for The sum of the angles of (n-2) triangles = (n-2)180^0
Sum of interior angles of a polygon with n sides = (n-2)180^0
= (2n-4)90^0
= (2n-4) right angles
EXERCISE:
If a polygon has 7 sides, it has……………. vertices
SUM OF EXTERIOR ANGLES OF A POLYGON
If the sides of a polygon are produced in order, the sum of exterior angles so
formed is always 360^0
3 3
2 4
2
4
1 5 1
5
[<1+<2+<3+<4+<5=360^0] [<1+<2+<3+<4+<5+<6=360^0]
7. Example 1: Is it possible to have a polygon, the sum of whose interior angles is
9 right angles.
Solution:
Let the number of sides be n.
There for sum of its interior angles = (2n-4)90^0
According to the given statement:
(2n-4)90^0=9*90^0 [9 right angles=9*90^0]
2n-4=9
n=6.5 which is not possible
EXERCISE:
One angle of a 7 sided polygon is 114^0 and each of the other 6 angles is
x^0. Find the magnitude of x ^0.
REGULAR POLYGON
A polygon said to be a regular polygon, if all its
(1) Interior angles are equal,
(2) Sides are equal and
(3) Exterior angles are equal.
(a) If a regular polygon has n sides:
8. 1. The sum of interior angles =(n-2)180^0
And, each interior angle = (n-2)180^0/n
2. The sum of its exterior angles =360^0
And, each exterior angle =360^0/n
3. Number of sides(n) of the regular polygon=360^0/Exterior angle
(b) Whether the giver polygon is regular or not, at each vertex of the
polygon:
Exterior angle + Interior angle=180^0
Since, both the angles together form a straight line angle
Example 1:
If each interior angle of a regular polygon is 144^0, find the number
of sides in it?
Solution:
Let the number of sides of the regular polygon is n
Its each interior angle=(2n-4)90^0/n i.e. 144^0=(2n-4)90^0/n
144 n=180 n-360 i.e. n=10
Number of sides=10
EXERCISE
The sum of the interior angles of a regular polygon is equal to six times
the sum of exterior angle. Find the number of sides of the polygon?
Is it possible to have a regular polygon with each interior angle is equal
to 105^0
