Question

find the area of shaded region of a circle of radius 14 cm if the length of the corresponding APB is 22cm​

Answer 1

Given :-

• Radius of circle = 14 cm
• length of corresponding arc = 22 cm

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To find :-

• Area of shaded region

Knowledge required :-

• Formula to find the measure of an arc

$\red{\bigstar}\boxed{\sf{Arc\:length=\dfrac{\theta}{360}\times 2\pi r}}$

• Formula to find the area of sector

$\red{\bigstar}\boxed{\sf{area\:of\:sector=\dfrac{\theta}{360}\times \pi r^2}}$

(where θ is the angle subtended by arc at centre and r is the radius of circle or sector)

Solution :-

Using formula for calculation of arc length

$\implies\sf{22=\dfrac{\angle APB}{360}\times 2 \; \pi \times 14}\\\\\\\implies\sf{22=\dfrac{\angle APB}{360}\times 2\times \dfrac{22}{7} \times 14}\\\\\\\implies\underline{\underline{\red{\sf{\angle APB = \dfrac{360}{4}=90^{\circ}}}}}$

Using formula for calculating area of sector

$\implies\sf{ar(sector\: APB)=\dfrac{90^{\circ}}{360^{\circ}}\times \pi \times (14)^2}\\\\\\\implies\sf{ar(sector\:APB)=\dfrac{1}{4}\times \dfrac{22}{7}\times 14 \times 14}\\\\\\\implies\underline{\underline{\red{\sf{ar(sector\:APB)=154\:cm^2}}}}$

Calculating area of triangle APB

$\implies\sf{ar(\triangle APB)=\dfrac{1}{2}\times 14\times 14}\\\\\\\implies\underline{\underline{\red{\sf{ar(\triangle APB)=98\;cm^2}}}}$

Now, calculating the area of shaded region

$\implies\sf{ar(shaded\:region)=ar(sector \:APB)\;-\;ar(\triangle APB)}\\\\\\\implies\sf{ar(shaded\:region)=154-98}\\\\\\\implies\overbrace{\underbrace{\large{\red{\underline{\underline{\sf{Ar(shaded\:region=56\:cm^2}}}}}}}$

Answer 2

Question:

Find the area of shaded region of a circle of radius 14cm if the length of the corresponding APB is 22 cm.

Answer:

The area of shaded region of the circle

: 56 cm²

Given :

➛Radius of the circle : 14 cm.

➛length of the corresponding APB is 22

cm.

To Find:

The area of shaded region of the circle .

Solution:

We are given,

➛Radius of the circle :( r) 14 cm.

➛length of the corresponding APB is 22

cm.

We know,

${\large{\green{\boxed{\pink{\star{\rm{Perimeter\:of\:the\:circle\: = 2 \pi r }}}}}}}$

$\implies \rm Perimeter\:of\:the\:circle\: = 2 \times \dfrac{22}{7} \times 14$

$\implies \rm Perimeter\:of\:the\:circle\: = 2 \times \dfrac{22}{ \cancel{7}} \times \cancel{14}(2)$

$\implies \rm Perimeter\:of\:the\:circle\: = 2 \times 22 \times 2$

${\therefore{\blue{\boxed{\orange{\star{\rm{Perimeter\:of\:the\:circle\: = 88 \: cm. }}}}}}}$

We also know,

${\large{\green{\boxed{\pink{\star{\rm{Area\:of\:the\:segment\: = \dfrac{Arc\:length}{Perimeter\:of\:the\:circle} \times Area\:of\:the\:Circle}}}}}}}$

but,

${\large{\green{\boxed{\pink{\star{\rm{Area\:of\:the\:circle\: = \pi {r}^{2} }}}}}}}$

$\implies \rm Area\:of\:the\:segment = \dfrac{22}{88} \times \dfrac{22}{7} \times {14}^{2}$

${\therefore{\blue{\boxed{\orange{\star{\rm{Area\:of\:the\:segment\: = 154 \: {cm}^{2} . }}}}}}}$

Now calculating the area of triangle APB

We know,

${\large{\green{\boxed{\pink{\star{\rm{Area\:of\:the\: \triangle \: = \dfrac{1}{2} \times base \times height }}}}}}}$

$\therefore \rm Area\:of\:the \: \triangle APB = \dfrac{1}{2} \times 14 \times 14$

$\therefore \rm Area\:of\:the \: \triangle APB = \dfrac{1}{\cancel{2}} \times \cancel{14} (2 ) \times 14$

${\therefore{\blue{\boxed{\orange{\star{\rm{Area\:of\:the \: \triangle APB = 98 \ {cm}^{2} . }}}}}}}$

Now,

${\small{\green{\boxed{\pink{\star{\rm{Area\:of\:the\:shaded\:region = Area\:of\:the\: segment - Area\:of\:the \: \triangle APB}}}}}}}$

$\therefore \rm Area\:of\:the\:shaded\:region = 154 - 98$

$\therefore \rm Area\:of\:the\:shaded\:region = 56 {cm}^{2}</p><p>$

${\therefore{\purple{\boxed{\red{\star{\rm{Area\:of\:the\:shaded\:region = 56 \ {cm}^{2} . }}}}}}}$