Question

$\bf \bold\blue{Question:-}$

The boundary of the shaded region consists of three semicircular arcs, the smaller ones being equal The diameter of larger arc is 10 cm.
Calculate
-> length of boundary
-> the area of shaded region
(Take π to be 3.14)

​

Answer 1

Answer:

• Length of boundary = 31.4 cm
• Area of Shaded Region = 39.25 cm².

Step-by-step explanation:

According to the Question

• Diameter of Larger arc = 10cm

Therefore , radius, r = 10/2 = 5cm

Also,

Diameter of smaller semicircle will be 5cm

Therefore , Radius of Semicircle, r' = 5/2 = 2.5cm

Formula Used :-

• Circumference of semicircle = πr
• Area of Semicircle = 1/2 πr²

Now , let's calculate

Length of boundary :-

From the given figure we observe that

Length of Boundary = Circumference of Lager Semicircular Arc + 2× Circumference of smaller semicircular Arc

by putting the value we get

⇢ Length of boundary = πr + 2 πr'

⇢ Length of boundary = π ( r + 2r')

⇢ Length of boundary = 3.14 ( 5 + 2×2.5) cm

⇢ Length of boundary = 3.14 ( 5+5) cm

⇢ Length of boundary = 3.14 (10) cm

⇢ Length of boundary = 31.4 cm

• Hence, the length of boundary is 31.4 cm .

Now calculating

Area of Shaded Region :

From the given figure we observe that

• Area of Shaded Region = Area of larger Semicircle

by putting the value we get

⇢ Area of Shaded Region = 1/2 πr²

⇢ Area of Shaded Region = 1/2 × 3.14 × 5²

⇢ Area of Shaded Region = 1.57 × 25

⇢ Area of Shaded Region = 39.25 cm²

• Hence, the area of shaded region is 39.25 cm².

Answer 2

$\Large \underline{ \underline{ \text{Question:}}}$

• The boundary of the shaded region consists of three semicircular arcs, the smaller ones being equal The diameter of larger arc is 10 cm. (Refer to Attachment).

Calculate

• > length of boundary
• > the area of shaded region

(Take π to be 3.14)

$\Large \underline{ \underline{ \text{Solution:}}}$

Given that,

• $\text{The Diameter of Larger Arc}{ = 10cm}$

As given in figure the Smaller arc is 1/2 of Larger arc,

Hence,

• $\text{The Diameter of Smaller Arc}{ = 5cm}$

As we know,

• $\boxed{ \text{Radius} = \frac{1}{2} \times \text{Diameter}}$

Finding the Radius (R) of Larger arc,

$\longrightarrow \text{Radius} \: (R) = \frac{1}{2} \times 10cm \\ \\ \longrightarrow \text{Radius} \: (R) = 5cm$

Finding the Radius (r) of Smaller arc,

$\longrightarrow \text{Radius} \: (r) = \frac{1}{2} \times 5cm \\ \\ \longrightarrow \text{Radius} \: (r) = 2.5cm$

Finding the Length of Boundary,

As given in figure,

$\longrightarrow \text{The Length}_{(\text{Boundary})}= \frac{1}{2} (2\pi R) + \frac{1}{2} (2\pi r) + \frac{1}{2} (2\pi r) \\ \\ \longrightarrow \text{The Length}_{(\text{Boundary})}= \pi (R + r+ r) \\ \\ \longrightarrow \text{The Length}_{(\text{Boundary})}= \pi(R + 2r)$

Substituting the values and Finding the Length of Boundary,

$\longrightarrow \text{The Length}_{(\text{Boundary})}= 3.14 [ 5cm + 2(2.5cm)] \\ \\ \longrightarrow \text{The Length}_{(\text{Boundary})}= 3.14 [ 5cm + 5cm] \\ \\ \longrightarrow \text{The Length}_{(\text{Boundary})}= 3.14 [ 10cm ] \\ \\ \longrightarrow \boxed{\text{The Length}_{(\text{Boundary})}= 31.4cm}$

Therefore,

• The Length boundary is 31.4cm.

.

Finding the Area of Shaded Region,

As given in figure,

$\longrightarrow \text{The Area}_{(\text{Shaded Region})}= \frac{1}{2} (\pi R^{2} ) - \frac{1}{2} (\pi {r}^{2} ) + \frac{1}{2} (\pi {r}^{2} ) \\ \\ \longrightarrow \text{The Area}_{(\text{Shaded Region})}= \frac{1}{2} \pi R ^{2}$

Substituting the values and Finding the Area Shaded Region,

$\longrightarrow \text{The Area}_{(\text{Shaded Region})}= \frac{1}{2} \pi R ^{2} \\ \\ \longrightarrow \text{The Area}_{(\text{Shaded Region})}= \frac{1}{2} (3.14) (5cm) ^{2} \\ \\ \longrightarrow \text{The Area}_{(\text{Shaded Region})}= \frac{1}{2} (3.14)(25cm ^{2}) \\ \\ \longrightarrow \text{The Area}_{(\text{Shaded Region})}= \frac{1}{2} (78.5 {cm}^{2} ) \\ \\ \longrightarrow \text{The Area}_{(\text{Shaded Region})}= 39.25 {cm}^{2}$

Therefore,

• The Area of Shaded Region is 39.25cm².

$\\ \Large \underline{ \underline{ \text{Required Answer:}}}$

• The Length boundary is 31.4cm.

• The Area of Shaded Region is 39.25cm².