Area of a part of a circle | Mathematics Form 2

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Area of a part of a circle - Mathematics Form 2
















Lesson objective

By the end of the lesson you should be able to:
Find the area of a common region between two circles.



INTRODUCTION

Trigonometric ratio:
      Given a right angled triangle, ABC.
Opposite1Hypotenus       A        
        Adjacent        
               

            
            B          C

Diagram 9
The longest side is called the Hypotenuse (AC) (Blink AC as it is read)
Side BC is Adjacent to angle ? and side AB is opposite to angle ?. (Blink AB and BC as they read)
Trigonometric ratios are stated with reference to angle ?


Sin of ? = 2

   
Cosine of ?= 4

Tangent of ? =5 

The ratios are abbreviated as Cos ?, Sino ? and Tan ?.
Area of sector


The following is a diagram showing a sector.


6

The arc subtend an angle of ?o at the centre of circle of radius r
The area of sector AOB is


      7


Note that the area of a sector is a fraction of the area of a circle which is 78r2.  The size of the fraction is determined by the angle of the sector eg;

  1. If the angle is ?; the fraction is 9
  2. If the angle is 60o, the fraction is 9

Example 1
Find the area of the following sector.

0
Solution:
Area of the sector = 89 
Area of the segment.

 

 

A sector is made up of a triangle (OAB) and a segment (the shaded part).
The area of the segment:

  1. Find the area of the sector.
  2. Find the area of triangle OAB.
  3. Subtract the area of the triangle OAB from the area of the sector.

Area of the sector =   89   
Area of the triangle = x r2 x sin?
Area of the segment = 0 
= 9 r2sin?.
Example 2
Find the area of the shaded part in the figure below. (? r = 3.142).

Diagram 13
 Solution:
The area of the sector is
45
The area of the triangle OPQ is
7 x 5 x 5 x sin60 = 67 x 25cm x 0.866
        =10.83cm2
Area of the segment (shaded part) is
13.09cm2 10.83cm2
      =2.26cm2.
Example 3

 

Diagram 14
Find the area of the segment in the following diagram given that XY is 12cm and radius is 10 cm.
Solution:

 

Diagram 15
Is perpendicular OT bisector of the chord XY?
XT=6cm and TY=6cm
Diagram 15
Using the right angled triangle OTY, find the height OT by Pythagoras theory.
 

Diagram 16
Height (H) = DY2 TY2
H = 102 62
H = 100-36 = 64
H=8cm.
Then:
Area of the triangle XOY
         = 5 x base x height
Base = 12cm.
Height = 8cm
Area = 6 x 12cm x 8cm =48cm2
To find the area of the sector, determine the value of ? which the sector angle.

Notice that OT bisects the angle ?.
Using the right angled TOY which is 34
We use the trigonometric ratios,


23


45


66


Therefore cosine 77  
To find the value of 45,  find the cosine invade of 0.8, Cos-1 0.8 = 24


Read off the value from the cosine tables


88    


Therefore the sector angle


? = 36.87 x 2 = 73.74
The area of the sector is


44


55

The area of the segment then = Area of the sector Area of triangle OXY
=64.36cm2 48cm2

=16.36cm2


Area of a common regions, between two circles.


When two circles interact they form a common region.


Oval:      A8 


Diagram 18
(i)  the following are two circles with centre A and B.

Diagram 19
Oval:      A88

(ii)  When the two circles intersect they form a common region.

Diagram 20

  1. When the circles are separated, we note that the common region is made of the two segments.

(Animate the two circles by letting them intersect to show the common region and then separate them to show the two segments).

The common area that is shaded is calculated by adding the areas of the two segments.

EXAMPLE 1
Find the area of the common shaded in diagram below where the circles have an equal reading of 7cm and is  
      77PAQ=77PBQ=600

Solution
Separate the circles to show the segments.

Diagram 22

To find the area of the common region
Find the area for the segment in the circle centre A


Area of the sector = 67


         =25.67cm2


Area of the triangle APQ
      = 4  x 7 x 7x sin60

      =21.22cm2


Area of the segment = 25.67-21.22
         =4.45cm2


Note that the two segments are equal.
Therefore the area of the common region is
      4.45cm x 2 = 8.9cm2

EXAMPLE 2
Find the area of the shaded region in the following diagram.

Solution

  1. To find the area of the sector in the circle centre P draw sector PUV.

  

   Diagram 24
   Find the area of triangle PUV
   7
To get 556 find the sin inverse 0.5= 30o.
Therefore PM = 10 Cos 30
       =10 x 0.866 = 8.66cm
Area of the triangle PVU = 34 x 10 x8.66
         =43.3 cm2
To find the area of the sector PUV, find the sector angle.
Sector angle is 2 x 30 =60o
79
      = 37cm2
Area of the segment
      = 52.37 43.3
      = 9.07 cm2

  1. To find the area of the section in circle centre Q. Draw sector

     

Find the area of the triangle
667
Sin-1 =0.7143=45.590
66= 45.59o
To find QM, use
Cos 45.59 = 667
QM = 4.899cm
Area of triangle UVQ
= 5  x 10 x 4.899
   =24.5cm2
To find the area of sector QUV, find the sector angle.
Sector angle = 45.59 x 2
      =91.81o
=    667
      =39cm2
Area of the segment
      =39 24.5
      =14.49
      =14.5 cm2
Area of the common region therefore is the sum of the, area of the segment in circle centre P = 9.07 cm
And
Area of the segment in circle centre
Q = 14.5cm
   Which is =9.07 + 14.5
=23.57cm2


(ii) A sector: This an area bounded by two radii and an arc.























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