Question

find the area of the shaded region for the given figure.​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Area\:of\:shaded\:region=112\:cm^{2}}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given :}} \\ \tt: \implies Side \: of \: square = 14 \: cm \\ \\ \red{\underline \bold{To \: Find :}} \\ \tt: \implies Area \: of \: shaded \: region =?$

• According to given question :

$\bold{In \: \triangle \: ABC} \\ \tt: \implies {AC}^{2} = {AB}^{2} + {BC}^{2} \\ \\ \tt: \implies {AC}^{2} = {14}^{2} + {14}^{2} \\ \\ \tt: \implies {AC}^{2} = 196 + 196 \\ \\ \tt: \implies {AC}^{2} =392 \\ \\ \tt: \implies {AC} = \sqrt{392} \\ \\ \green{\tt: \implies {AC} =14 \sqrt{2} \: cm} \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies Area \: of \: shaded \: region = Area \: of \: quadrant - Area \: of \: square \\ \\ \tt: \implies Area \: of \: shaded \: region = \frac{1}{4}\pi {r}^{2} - {side}^{2} \\ \\ \tt: \implies Area \: of \: shaded \: region = \frac{1}{4} \times \frac{22}{7} \times {(14 \sqrt{2} )}^{2} - {14}^{2} \\ \\ \tt: \implies Area \: of \: shaded \: region = \frac{22}{28} \times 392 - 196 \\ \\ \tt: \implies Area \: of \: shaded \: region =22 \times 14 - 196 \\ \\ \tt: \implies Area \: of \: shaded \: region =308 - 196 \\ \\ \green{\tt: \implies Area \: of \: shaded \: region =112 { \: cm}^{2} }$

Answer 2

$\red{\underline{\underline{Answer:}}}$

$\sf{Area \ of \ shaded \ region \ is \ 112 \ cm^{2}}$

$\sf\orange{Given:}$

$\sf{For \ square,}$

$\sf{\implies{Side=14 \ cm}}$

$\sf\pink{To \ find}$

$\sf{Area \ of \ the \ shaded \ region.}$

$\sf\green{\underline{\underline{Solution:}}}$

$\sf{For \ square,}$

$\boxed{\sf{Area \ of \ square=side^{2}}}$

$\sf{Area \ of \ square=14^{2}}$

$\sf{\therefore{Area \ of \ square=196 \ cm^{2}...(1)}}$

$\boxed{\sf{Diagonal \ of \ square=Side\times\sqrt2}}$

$\sf{\therefore{Diagonal \ of \ square=14\sqrt2 \ cm}}$

$\sf{For \ sector,}$

$\sf{\theta=90°...All \ angles \ of \ square \ are \ 90°}$

$\sf{Radius (r)=Diagonal \ of \ square.}$

$\sf{\therefore{Radius (r)=14\sqrt2 \ cm}}$

$\boxed{\sf{Area \ of \ sector=\frac{\theta}{360}\times\pi\times \ r^{2}}}$

$\sf{\therefore{Area \ of \ sector=\frac{90}{360}\times\frac{22}{7}\times(14\sqrt2)^{2}}}$

$\sf{\therefore{Area \ of \ sector=22\times14}}$

$\sf{\therefore{Area \ of \ sector=308 \ cm^{2}...(2)}}$

$\sf{Area \ of \ shaded \ region}$

$\sf{=Area \ of \ sector \ - \ Area \ of \ square}$

$\sf{...from \ (1) \ and \ (2)}$

$\sf{=308-196}$

$\sf{\therefore{Area \ of \ shaded \ region=112 \ cm^{2}}}$

$\sf\purple{\tt{\therefore{Area \ of \ shaded \ region \ is \ 112 \ cm^{2}}}}$