Circle I: Introduction of a Circle & its Properties – SS2 Further Mathematics Lesson Note (Second Term)

Introduction of a circle and its properties 

1. Parts Of A Circle: 

– The centre is the point in the middle of a circle. 

– The circumference is the curved outer boundary of the circle. 

– An arc is a curved part of the circumference. 

– A radius is any straight line joining the centre to the circumference. The plural of radius is radii.

– A chord is any straight line joining two points on the circumference. 

– A diameter is a straight line which divides the circle into two equal parts or a diameter is any chord which goes through the center of the circle.

ii. Region Of A Circle

– A sector is the region between two radii and the circumference. 

– A semicircle is a region between a diameter and the circumference i.e. half of the circle. 

– A segment is the region between a chord and the circumference.

Given a circle center O with radius r. The circumference of the circle is 2Пr. Therefore, the length, L, of arc XY is given as L =   θ x  2Пr ÷ 360°

Where θ is the angle subtended at the center by arc XY and r is the radius of the circle.

Also,

The perimeter of Sector XOY = r + r + L

Where L = length of arc XY 

= θ     X    2 Пr ÷ 360

 Then Perimeter of Sector XOY 

=r  +  r  + L

= 2r  +   θ     x    2 Пr ÷ 360°

EXAMPLES

1. An arc of length 28cm subtends an angle of 240°  at the centre of a circle. In the same circle, what angle does an arc of length 35cm subtend?

 Calculate the perimeter of a sector of a circle of radius 7cm, the angle of the sector being 108°, if П is 22/7.

Solutions:

1. L =    θ   x    2 Пr ÷ 360°

When L = 28cm ,  θ =  240, r = ?       

Then    L =  θ    x    2 Пr ÷ 360°

 28 = 24   x    2 x (22 ÷ 7)  x r   ÷   360°            

Cross-multiply:

24  x  44 x r  =  28 x  360 x  7

15.                 7         60

r = 28  x  360  x  7 cm

              24×44

       4         11

 r  =  49  x   15  cm

                11

r = 735 cm

      11

Also When L = 35cm, r = 735 cm

                                     11θ  = ?

 Then 

L =  θ   x  2 Пr

           360°

 35 = θ   x 2  x  22  x  735

      360              7        11

 Then,

Cross multiply

x  360 x 7 x 11 = θ x 44 x 735

   1                                  11

35 x 360 x 77    =  θ

 44     x       735

   4        105  3

360    =   θ  = 30°

12

Thus, when the length of the arc is 35cm, the angle subtended at the centre is 30°

2. Perimeter of a sector of a circle = 2r + θ   x 2 Пr ÷ 360°

=  2 x 7 + 108   x 2 x 22 x 7

            360             7  1

          3

= 14 +  108  x  44cm

    360 10

= 14 + 3 x 44 cm

              10

=  14 + 132 cm

             10 

=  14 + 13.2 cm

= 27.2 cm