CLASS-9
CIRCUMFERENCE & AREA OF A CIRCLE - INTRODUCTION & FORMULAE

CIRCUMFERENCE & AREA OF CIRCLE –

For every circle circle, the ratio between the circumference and diameter in a constant. This constant is represented by a Greek letter π (Pi)

              Circumference

So,  π = -----------------

                Diameter

1. For a circle of radius = r units, we have -

(i)  Circumference of the circle

      = 2πr units                                = πd units

Where d is the parameter, and d = 2r

(ii) Area of a circle = πr² sq.units




2.  For a semi circle of radius = r units, we have – 

                                        1

(i)        Area of the semi-circle = ------- πr²  sq.units

                                        2

(ii)          Perimeter of a semi-circle = (πr + 2r) units



3. Area of a Circular Ring –

If R & r be the outer and inner radii of a ring, then

Area of the ring = π (R²- r²) sq.units







4. Results on Sectors & Segments –

Suppose an arc ACB makes an angle θᵒ at the center O of a circle of







                                    2πrθ

(i)                 Length of arc ACB =   (---------) units

                                     360

                                                

                                      πr²θ

(ii)               Area of Sector OACBO = (-------) units

                                       360

                          1               2πrθ

                     = ------ X r X (--------) sq.units

                          2                360

                           1

                     = (------ X radius X arc length) sq.units

                           2

(iii)    Perimeter of the Sector OACBO= length of the arc ACB + OA + OB

                                             2πrθ

                                       =  (------- + 2r) units

                                              360



5. Rotation Made by an wheel –

(i)  Distance moved by a wheel in 1 revolution

                                             = Circumference of the Wheel

(ii)  Number of rotations made by a wheel in unit time

         Distance moved by it in unit time

   = --------------------------------

            Circumference of the wheel



6. Facts about Clocks :

(i)      Angle described by minute hand in 60 minutes = 360ᵒ

                                                          360

(ii)    Angle described by minute hand in 5 minutes = (------ X 5) = 30ᵒ

                                                           60

(iii)    Angle described by hour hand in 12 hours = 360ᵒ

Angle described by hour hand in 1 hour = 30ᵒ



7. Equilateral Triangle -

In a equilateral triangle of side a units, we have

                                     √3a

(i)     Height of the triangle, h = ------ units

                                      2

                                √3a²

(ii)     Area of the Triangle = (------) sq.units

                                   4

(iii)     Radius of incircle, r = h/3 = (a/2√3) units

(iv)     Radius of circumcircle, R = 2h/3 = (a/√3) units

Thus, r = a/2√3 and R = a/√3