Question

Find the area and the perimeter of the shaded region in figurer
The diamensions are in centimetres
​

Answer 1

$\underline{\large{\sf\ \ \ \ \ SOLUTION:-\ \ \ \ }}$

• Radius of Big Semicircle(R) = 14cm

• Radius of Small semicircle(r) = 14/2= 7cm

We have to find the perimeter and the area of shaded region of the shaded region

First find the Area of Shaded region :-

$\begin{gathered}\star\sf\ \ Area\ of\ shaded\ region = \big(Area\ of\ Big_{Semicircle}\big)-\big(Area\ of\ Small_{semicircle}\big)\\ \\ \\ \star\sf\ \ Area\ of\ semicircle= \dfrac{\pi r^2}{2}\\ \\ \\ :\implies\sf\ Area\ of\ shaded\ region = \dfrac{\pi R^2}{2}-\dfrac{\pi r^2}{2}\\ \\ \\ :\implies\sf\ Area\ = \dfrac{\pi }{2}\Big(R^2-r^2\Big)\\ \\ \\:\implies\sf\ Area\ = \dfrac{\cancel{22}}{7\times \cancel{2}}\times \Big\{(R+r)(R-r)\Big\}\\ \\ \\ :\implies\sf \ Area\ = \dfrac{11}{7}\times \Big\{(14+7)(14-7)\Big\}\\ \\ \\ :\implies\sf\ Area\ = \dfrac{11}{\not{7}}\times 21\times \not{7}\\ \\ \\ :\implies\sf\ Area\ = 11\times 21\\ \\ \\ :\implies\sf\ Area\ = 231cm^2\\ \\ \\ \underline{\pink{\sf \ Area\ of\ shaded\ Region= 231cm^2}}\end{gathered}$

Now the perimeter of shaded region :-

Perimeter= Circumference of big semicircle + Circumference of small semicircle + Radius of big semi Circle.

$\begin{gathered}\bigstar\sf\ \ \ Perimeter\ of\ semicircle = \dfrac{2\pi r}{2}\\ \\ \\ \dashrightarrow\sf\ Perimeter\ of\ semicircle= \pi r\\ \\ \\ :\implies\sf\ Perimeter= (\pi R+ \pi r)+ R\\ \\ \\ :\implies\sf\ \ Perimeter= \pi(R+r)+R\\ \\ \\ :\implies\sf\ \ Perimeter= \dfrac{22}{7}\times \Big(14+7\Big)+ 14\\ \\ \\ :\implies\sf\ \ Perimeter= \Bigg\{\dfrac{22}{\cancel{7}}\times \cancel{21} \Bigg\}+14\\ \\ \\ :\implies\sf\ \ Perimeter= \big(22\times 3\big)+14\\ \\ \\ :\implies\sf\ \ Perimeter= 66+14\\ \\ \\ :\implies\sf\ Perimeter= 80cm\\ \\ \\ \underline{\bigstar{\textsf{\textbf{Perimeter\ of\ shaded\ region= 80cm}}}}\end{gathered}$

Answer 2

Answer:

\underline{\large{\sf\ \ \ \ \ SOLUTION:-\ \ \ \ }}

SOLUTION:−

Radius of Big Semicircle(R) = 14cm

Radius of Small semicircle(r) = 14/2= 7cm

We have to find the perimeter and the area of shaded region of the shaded region

First find the Area of Shaded region :-

\begin{gathered}\begin{gathered}\star\sf\ \ Area\ of\ shaded\ region = \big(Area\ of\ Big_{Semicircle}\big)-\big(Area\ of\ Small_{semicircle}\big)\\ \\ \\ \star\sf\ \ Area\ of\ semicircle= \dfrac{\pi r^2}{2}\\ \\ \\ :\implies\sf\ Area\ of\ shaded\ region = \dfrac{\pi R^2}{2}-\dfrac{\pi r^2}{2}\\ \\ \\ :\implies\sf\ Area\ = \dfrac{\pi }{2}\Big(R^2-r^2\Big)\\ \\ \\:\implies\sf\ Area\ = \dfrac{\cancel{22}}{7\times \cancel{2}}\times \Big\{(R+r)(R-r)\Big\}\\ \\ \\ :\implies\sf \ Area\ = \dfrac{11}{7}\times \Big\{(14+7)(14-7)\Big\}\\ \\ \\ :\implies\sf\ Area\ = \dfrac{11}{\not{7}}\times 21\times \not{7}\\ \\ \\ :\implies\sf\ Area\ = 11\times 21\\ \\ \\ :\implies\sf\ Area\ = 231cm^2\\ \\ \\ \underline{\pink{\sf \ Area\ of\ shaded\ Region= 231cm^2}}\end{gathered}\end{gathered}

⋆ Area of shaded region=(Area of Big

Semicircle

)−(Area of Small

semicircle

)

⋆ Area of semicircle=

2

πr

2

:⟹ Area of shaded region=

2

πR

2

−

2

πr

2

:⟹ Area =

2

π

(R

2

−r

2

)

:⟹ Area =

7×

2

22

×{(R+r)(R−r)}

:⟹ Area =

7

11

×{(14+7)(14−7)}

:⟹ Area =



7

11

×21×



7

:⟹ Area =11×21

:⟹ Area =231cm

2

Area of shaded Region=231cm

2

Now the perimeter of shaded region :-

Perimeter= Circumference of big semicircle + Circumference of small semicircle + Radius of big semi Circle.

\begin{gathered}\begin{gathered}\bigstar\sf\ \ \ Perimeter\ of\ semicircle = \dfrac{2\pi r}{2}\\ \\ \\ \dashrightarrow\sf\ Perimeter\ of\ semicircle= \pi r\\ \\ \\ :\implies\sf\ Perimeter= (\pi R+ \pi r)+ R\\ \\ \\ :\implies\sf\ \ Perimeter= \pi(R+r)+R\\ \\ \\ :\implies\sf\ \ Perimeter= \dfrac{22}{7}\times \Big(14+7\Big)+ 14\\ \\ \\ :\implies\sf\ \ Perimeter= \Bigg\{\dfrac{22}{\cancel{7}}\times \cancel{21} \Bigg\}+14\\ \\ \\ :\implies\sf\ \ Perimeter= \big(22\times 3\big)+14\\ \\ \\ :\implies\sf\ \ Perimeter= 66+14\\ \\ \\ :\implies\sf\ Perimeter= 80cm\\ \\ \\ \underline{\bigstar{\textsf{\textbf{Perimeter\ of\ shaded\ region= 80cm}}}}\end{gathered}\end{gathered}

★ Perimeter of semicircle=

2

2πr

⇢ Perimeter of semicircle=πr

:⟹ Perimeter=(πR+πr)+R

:⟹ Perimeter=π(R+r)+R

:⟹ Perimeter=

7

22

×(14+7)+14

:⟹ Perimeter={

7

22

×

21

}+14

:⟹ Perimeter=(22×3)+14

:⟹ Perimeter=66+14

:⟹ Perimeter=80cm

★Perimeter of shaded region= 80cm