Question

17. In the given figure, find the area of the
shaded region where O is the centre of the
circle of radius 14 cm and circular arcs are
drawn with centres at A, B, C, D, E and F
respectively and each with radius 14 cm.​

Answer 1

Given :   circle of radius 14 cm .   circular arcs are  drawn with centres at A, B, C, D, E and F  . Respectively and each with radius 14 cm.​

To find : area of the  shaded region

Solution:

OA = PB = OC = OD = OE = OF = 14 cm  Radius

AB = BC = CD = DE = EF = FA = 14 cm    

All 6 shades regions are congruent

so lets find area of one shaded region

ΔOAB will be equilateral triangle with all sides = 14 cm

arc AOB will have 60° angle

Hence area of half of OB = 60° arc area - ΔOAB area

= (60/360)π 14² - (√3 /4 )14²

= (1/6)(22/7) 14² - 49√3

= 308/3  -  49√3

area of  Shaded OB =   2 ( 308/3  -  49√3)

there are 6 of such type

Hence total area of shaded region = 6 * 2 ( 308/3  -  49√3)

= 1232 - 1018.45

= 213.55 cm²

total area of shaded region =  213.55 cm²

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