This document contains information about perimeter, circles, arcs, angles, and areas in mathematics. It defines perimeter as the distance around a closed figure and explains how to find the perimeter of squares and circles. It also introduces pi (π) as the ratio of a circle's circumference to its diameter. The document discusses what arcs and sectors are, how to calculate the length of an arc using central angles, and how to find the area of sectors as a proportion of the whole circular area based on the central angle.

1. MATHEMATICS
STANDARD
IX
Government of Kerala
DEPARTMENT OF EDUCATION

2. NATIONAL ANTHEM
Jana-gana-mana-adhinayaka,jaya he
Bharata-bhagya-vidhata.
Punjab-Sindh-Gujarat-Maratha
Dravida-Utkala-Banga
Vindhya-Himachala-Yamuna-Ganga
Uchchala –Jaladhi-taranga.
Tava shubha asisa jage,
Tava subha asisa mage,
Gahe tava jaya gatha,
Jana-gana-mangala-dayaka jaya he
Bharata-bhagya-vidhata.
Jaya he, jaya he, jaya he,
Jaya jaya jaya , jaya he!
PLEDGE
India is my country. All Indians are my brothers and sisters. I love my country, and I am
proud of its rich and varied heritage. I shall always strive to be worthy of it. I shall give
respect to my parents, teachers and all elders and treat everyone with courtesy. I
pledge my devotion to my country and my people. In their well-being and prosperity
alone lies my happiness

3. CONTENTS
 Perimeter
 Perimeter and diameter
 Circles and polygons
 A new number
 Arc
 Arcs and angles
 Length of an arc
 Area
 Sectors

4. Chapter - 11
Circular Measures

5. Perimeter
The perimeter is the sum of the length
of all sides of a closed figure.
What is the perimeter of a square of
side 3cm?
The perimeter is 3+3+3+3=12cm.

6. How do we find the perimeter of a circle of
diameter 3cm?
We cannot compute it as in the case of
a square;
We can place a string around it
,straighten and measure.

7. perimeter and diameter
When the diameter is increased,the
perimeter also increases.

8. Circles and polygons

9. A new number
 The perimeter of a circle is proportional
to its diameter.
 The perimeter of any circle divided by it
diameter must give the same number.
 Actually this number is irrational . In fact
there is a special symbol in mathematics
for this number . This number pi.
perimeter of circle
diameter of circle

10. Arcs
An arc is a portion of circle.
AB and PQ are parts of a circle.
Usually ,we write AB or PQ to denote
the line joining two points.

11. Arcs and angles
In the figure below , ABP is an arc of the circle.
Suppose the point P moves away from A,along
the circle.
A p A
A

12. Then the length of the arc ABP also increases.
A
B
 P

13. Central angle
The angle made by joining the end
points of an arc to the centre of the
circle is called the central angle of the
arc.

15. Length of an arc
If the radius of a circle is denoted by r,its
perimeter is 2 r.So the length of an arc of
central angle 1 is
1360 of perimeter=2 r*1360
arc length=2360 of perimeter=2 r*2360
For an arc of central angle 12,
Arc length=12*1360 of perimeter=2
r*12*1360.
In general,for an arc of central angle x.
2 r*x*1360=2 r*x360.

16. The length of an arc of a circle is that part of
the perimeter of the circle,as the central
angle is of 360.

17. Area

19. By joining the vertices of the polygon to the center
of the circle , we can divide the polygon into
triangle

20. ‘

22. Sectors
In the figure below,two points of a circle
are joined to the centre.
The figure obtained thus is called a
sector of the circle.
Thus a sector is formed by an arc of a
circle and the radii through its end
points.

23. LOOK AT THE PICTURE
As the central angle increases, so does
the area of the sector. We can show
that the area of a sector of central
angle x is x/360 of the area of the
whole circle.

24. The area of a sector of a circle is
that part of the area of the circle
as the central angle is of 360
degree.

25. Area of a sector
In a circle of radius r, a sector of
central angle x has Area,
πr2 x x
360

26. THANK YOU