Q. 173.5( 6 Votes )

Let us write perimeter and area of circular shaded sector below.


Answer :

(i) Given: Radius of the circle = 12 cm and angle subtended by the arc = 90°


We know that the length of the arc


Length of arc AB


Length of AB:


∵ ∠ O = 90°


Using Pythagoras theorem,


OA2 +OB2 = AB2


122 + 122 = AB2


AB2 = 288


AB=12√2 cm


AB =16.92 cm


Perimeter of the circular shaded sector = Length of arc AB + length of AB


Perimeter of the circular shaded sector=18.857 + 16.92 =35.777 cm


Now, Area of the segment AB = area of sector ABO – area of triangle ABO


We know that the area of the minor sector


Area of ABO


Area of ABO = 113.14 cm2


In ∆ABO,


O = 90°, AO = BO = 12 cm {radius of the circle}




Area of ∆ABO = 72 cm2


Area of the segment AB = area of sector ABO – area of triangle ABO


Area of the segment AB = 113.14 – 72


Area of the segment AB = 41.14 cm2


(ii) Given: Radius of the circle = 42 cm and angle subtended by the arc = 60°


We know that the length of the arc


Length of arc


Length of AC:


In ∆ABC,


B = 60°, AB = BC = 42 cm {radius of the circle}


⇒∠ ABC = ACB {angles opposite to equal sides are equal}


By the angle sum property of the triangle,


BAC + ACB + B = 180°


2 BAC = 180° - 60°


⇒∠ BAC = 60°


Hence, ∆ ABC is an equilateral triangle.


AC = 42 cm


Perimeter of the circular shaded sector = Length of arc AC + length of AC


Perimeter of the circular shaded sector = 42 + 44 = 86 cm


Now, Area of the segment AC = area of sector ACB – area of triangle ACB


We know that the area of the minor sector


Area of ACB =


Area of ACB = 924 cm2


We know that


Area of a equilateral triangle where a is the side of it.



Area of ∆ABC = 763.83cm2


Area of the segment AC = area of sector ABC – area of triangle ABC


Area of the segment AC = 924 – 763.83


Area of the segment AC = 160.17 cm2


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