We know that : If in a Circle of Radius 'r', an Arc of length 'l' subtends an Angle 'θ' radian at the Centre then Relation between them is given by :
✿ [tex]\mathsf{\theta = \frac{l}{r}}[/tex]
Given : The Arc Length of the Larger Sector = 12 m
Given : θ = 1 radian
[tex]\mathsf{\implies 1 = \frac{12}{Radius\;of\;the\;Larger\;Sector}}[/tex]
[tex]\mathsf{\implies Radius\;of\;the\;Larger\;Sector = 12m}[/tex]
Given : The Arc Length of the Smaller Sector = 8 m
[tex]\mathsf{\implies 1 = \frac{8}{Radius\;of\;the\;Smaller\;Sector}}[/tex]
[tex]\mathsf{\implies Radius\;of\;the\;Smaller\;Sector = 8m}[/tex]
The Way to look at this Solution is to Realize that : If we Subtract the Area of Smaller Sector from the Area of Larger Sector, we will end up with the Area of the Shaded Region.
We know that : Area of a Sector with Radius r is given by : [tex]\mathsf{\frac{1}{2}(r)^2\theta}[/tex]
where : θ is the Angle subtended (in Radians)
[tex]\mathsf{\implies Area\;of\;the\;Larger\;Sector = \frac{1}{2}(12)^2(1)}[/tex]
[tex]\mathsf{\implies Area\;of\;the\;Larger\;Sector = \frac{144}{2}}[/tex]
[tex]\mathsf{\implies Area\;of\;the\;Larger\;Sector = 72m^2}[/tex]
[tex]\mathsf{\implies Area\;of\;the\;Smaller\;Sector = \frac{1}{2}(8)^2(1)}[/tex]
[tex]\mathsf{\implies Area\;of\;the\;Smaller\;Sector = \frac{64}{2}}[/tex]
[tex]\mathsf{\implies Area\;of\;the\;Smaller\;Sector = 32m^2}[/tex]
Area of the Shaded Region = (72 - 32) = 40m²
