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Find the perimeter and area of the shaded region.
"
 Given :
AD = 6 cm ; AB = BC = CD
To Find :
i) Perimeter of shaded region
ii) Area of shaded region
Solution :
AD = 6 cm ; AB = BC = CD
$AD \ = \ AB + BC + CD$
$AD \ = \ AB + AB + AB$
$AD \ = \ 3 AB$
$ \begin{array}{l}
\frac{AD}{3} \ =\ AB\\
\\
AB\ \ =\ \ \frac{6}{3}
\end{array}$
AB = 2 cm ; BC = 2 cm ; CD = 2 cm
There are Three Circles.
Circle 1 :
Diameter AD = 6 cm
$ \begin{array}{l}
Radius\ =\ \frac{Diameter}{2}\\
\\
Radius\ =\ \frac{6}{2}
\end{array}$
Radius (r1)= 3 cm
Circle 2 :
Diameter BD = 4 cm
Radius (r2) = BC = 2 cm
Circle 3 :
Diameter AB = 2 cm
$Radius\ =\ \frac{Diameter}{2}$
$Radius\ =\ \frac{2}{2}$
Radius (r3)= 1 cm
i) Perimeter of Shaded region :
Formula to find Circumference of circle = 2πr
Circumference of semi circle = $\frac{2πr}{2}$
Circumference of semi circle = πr
Perimeter of Shaded region = Circumference of semi circle 1 + Circumference of
semi circle 2 + Circumference of semi circle 3
Perimeter of Shaded region = $$\displaystyle π\ r\ _{1\ } \ +\ π\ r\ _{2} \ +\ π\ r\ _{3}$$
Take π as common ,
$$\displaystyle π\ ( \ r_{1} \ +r_{2} \ +r_{3} \ \ )$$
$$\displaystyle π\ ( 3\ +\ 2\ +\ 1)$$
π 6
Perimeter of Shaded region = 6 π cm
ii) Area of shaded region :
Formula to area of circle = πr 2
Area of semi circle = $\frac{π\ r^{2}}{2}$
Area of Shaded region = Area of semi circle 1 - Area of semi circle 2 + Area of semi
circle 3
Area of Shaded region =$$\displaystyle \frac{1}{2}\left( \ π\ r\ ^{2}_{1} -\ π\ r\ ^{2}_{2} \ +\ π\ r\ ^{2}_{3} \ \right)$$
Take π as common,
$$\displaystyle \frac{π}{2}\left( \ \ r\ ^{2}_{1} -\ \ r\ ^{2}_{2} \ +\ \ r\ ^{2}_{3} \ \right)$$
$$\displaystyle \frac{π}{2} \ \left( \ 3\ ^{2}_{\ } \ -2\ ^{2}_{\ } \ +1\ ^{2} \ \ \right)$$
$$\displaystyle \frac{π}{2} \ ( 9\ -\ 4\ +\ 1)$$
$$\displaystyle \frac{6\ π}{2}$$
Area of Shaded region = 3 π cm2
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