# CLASS-9CIRCUMFERENCE & 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

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

360

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

60

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

Angle described by hour hand in 1 hour = 30

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/23) units

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

Thus, r = a/23 and R = a/√3