Question

The adjecant figure consists of four small semi-circles of equal radius and two big semi-circles of equal radius (each 42cm). Find the area of the shaded region.​

Answer 1

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

${\tt{\therefore{Area\:of\:shaded\:region=82.6875\pi\:m^2=259.875\:m^2}}}$

$\sf{Given : } \\ \sf \implies \: Diameter \: of \: small \: semi \: circle (r_{2})= 10.5 \: m \\ \\ \sf \implies \: Diameter \: of \: big \: semi \: circle( r_{1})= 21\: m \\ \\ \sf{To \: Find : } \\ \sf\implies Area \: of \: shaded \: region =?$

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

$\implies r_{1} = \frac{21}{2} = 10.5 \: m \\ \\ \implies r_{2} = \frac{10.5}{2} = 5.25\: m \\ \\ \bold{Using \: formula \: of \: semi \: circle(For \: big \: semi \: circle)} \\ \sf \implies Area \: of \: 2 \: semi \: circle = 2 \times \frac{\pi {r}^{2} }{2} \\ \\ \sf \implies Area \: of \: 2 \: semi \: circle = \pi \times {10.5}^{2} \\ \\ \sf \implies Area \: of \: 2 \: semi \: circle =110.25\pi { \: m}^{2} \\ \\ \bold{Similarly(For \: small \: semi \: circle)} \\ \sf \implies Area \: of \: 2 \: \: semi \: circle =2 \times \frac{\pi}{2} { r_{2}}^{2} \\ \\ \sf \implies Area \: of \: 2 \: semi \: circle =\pi \times {5.25}^{2} \\ \\ \sf \implies Area \: of \: 2 \: semi \: circle =27.5625\pi \: {m}^{2} \\ \\ \bold{For \: area \: of \: shaded \: region} \\ \sf \implies Area \: of \: shaded \: region =Area \: of \: (big - small)semi \: circle \\ \\ \sf \implies Area \: of \: shaded \: region =110.25\pi - 27.5625\pi \\ \\ \sf \implies Area \: of \: shaded \: region =82.6875 \pi\: {m}^{2}=259.875\:m^2$

Answer 2

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

$\green{\tt{\therefore{Area\:of\:shaded\:region=259.875\:m^2}}}$

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

$\green{\underline{\bold{Given :}}} \\ \tt: \implies Diameter \: of \: big \: semi -circle(r_{1}) = 21\: m \\ \\ \tt: \implies Diameter \: of \: small\:semi- circle (r_{2})= 10.5\: m \\ \\ \red{\underline{\bold{To\:Find :}}} \\ \tt: \implies Area \: of \: shaded \: region =?$

• According to given question :

$\green{\star}\tt\: r_{1} = \frac{21}{2} = 10.5 \: m \\ \\ \green{\star}\tt\: r_{2} = \frac{10.5}{2} = 5.25\: m \\ \\ \bold{As\:we\:know\:that} \\ \tt: \implies Area \: of \: big\:circle = 2 \pi r_{1}^{2} \\ \\ \tt: \implies Area \: of \: big\:circle = \frac{22}{7} \times {10.5}^{2} \\ \\ \tt: \implies Area \: of \: big\:circle = 346.5{ \: m}^{2} \\ \\ \bold{Similarly:} \\ \tt: \implies Area \: of \: small\:circle = \frac{\pi}{2} { r_{2}}^{2} \\ \\ \tt: \implies Area \: of \: small\:circle = \frac{22}{7} \times {5.25}^{2} \\ \\ \tt: \implies Area \: of \: small\:circle = 86.625\:m^2 \\ \\ \bold{For \: area \: of \: shaded \: region} \\ \tt:\implies Area \: of \: shaded \: region = Area \: of \: (big\:circle - small \: circle) \\ \\ \tt:\implies Area \: of \: shaded \: region =346.5 - 86.625 \\ \\ \tt:\implies Area \: of \: shaded \: region =259.875\: {m}^{2}$