MATHEMATICS

STANDARD

IX

Government of Kerala

DEPARTMENT OF EDUCATION

NATIONAL ANTHEM

Jana-gana-mana-adhinayaka,jaya heBharata-bhagya-vidhata.Punjab-Sindh-Gujarat-MarathaDravida-Utkala-BangaVindhya-Himachala-Yamuna-GangaUchchala –Jaladhi-taranga.Tava shubha asisa jage,Tava subha asisa mage,Gahe tava jaya gatha,Jana-gana-mangala-dayaka jaya heBharata-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

CONTENTS

PerimeterPerimeter and diameterCircles and polygonsA new numberArcArcs and anglesLength of an arcAreaSectors

Chapter - 11

Circular Measures

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.

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.

perimeter and diameter

When the diameter is increased,the perimeter also increases.

Circles and polygons

A new numberThe 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

Arcs

An arc is a portion of circle. P QA BAB and PQ are parts of a circle.Usually ,we write AB or PQ to

denote the line joining two points.

Arcs and angles

In the figure below , ABP is an arc of the circle. A PSuppose the point P moves away from A,along

the circle. A p A

A

B

B

B

Then the length of the arc ABP also increases.

A B P

O O

A

BP

As the length of the arc ABP increases , so does the angle AOP At the center of the circle

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.

AB

P

O A

B

P

O

O

A

B

P

The central angle of the arc ABP is now 180 degree .suppose P goes down45 degreeMore , the central angle of ABP is 225 degree. When P moves 45 degree more to the Right the central angle becomes 270 degree . Another 45 degree up and it becomes

315 degree.45 degree more upwards and ABP becomes the full circle.

Length of an arcIf 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

1\360 of perimeter=2 r*1\360 arc length=2\360 of perimeter=2

r*2\360For an arc of central angle 1\2,Arc length=1\2*1\360 of perimeter=2

r*1\2*1\360.In general,for an arc of central angle x. 2 r*x*1\360=2 r*x\360.

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

Area

Draw regular polygons with more and more sides with in the circle , their areas would get

closer and closer to the area of the circle.

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

If the polygon has n sides, then its area is :n*(1/2)sh= (1/2)nsh‘ns’ is the sum of all sides of the polygon

In other words ns is perimeter

Perimeter denoted by p

Area of the polygon = (1/2)ph

Increase the number of sides , the polygon gets closer and closer to the circle . The perimeter and area of the polygon get closer and closer to the

perimeter and area of the circle

Area of the circle = (1/2)*perimeter of the circle * radius

= (1/2)*2∏*radius*radius

= ∏*square of radius

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.

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.

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.

Area of a sector

In a circle of radius r, a

sector of central angle x has Area,

πr2 x x

360

THANK YOU