Question

the circumference of two concentric circles are 132 cm and 88 cm respectively. Find the area of ring enclosed between them.​

Answer 1

${\huge{\underline{\bf Solution:}}}\\ \\ \\ {\underline{\bf Given:}}\\ \\ \implies \sf Circumference\;of\;1st\;circle = 88\;cm\\ \\ \implies \sf Circumference;of\;2nd\;circle=132\;cm\\ \\ \\{\underline{\bf To\;find:}}\\ \\ \implies \sf Area\;of\;Ring\\ \\ \\{\underline{\bf Formula\;used:}}\\ \\ \implies \sf Circumference\;of\;circle=2\pi r\\ \\ \implies \sf Area\;of\;circle=\pi r^{2}\\ \\ \rule{200}{1}$

${\sf{\underline{Let,\;radius\;of\;1st\;circle\;be\;'r'}}\\ \\{\sf \underline{\;and\;radius\;of\;2nd\;circle\;be\;'R'.}}}$

${\underline{\sf Now,\;we\;will\;calculate\;radius\;by\;circumference\;formula,}}\\ \\ \\ \implies \sf Circumference\;of\;1st\;circle=2\pi r\\ \\ \implies \sf 88 = 2\times 3.14\times r\\ \\ \implies 88=6.28\times r\\ \\ \implies \sf r=\dfrac{88}{6.28}\\ \\ \implies \sf r=14\;cm\;(Approx)\\ \\ \rule{200}{1}\\ \\ \implies \sf Circumference\;of\;2nd\;circle=2\pi R\\ \\ \implies \sf 132 = 2\times 3.14\times R\\ \\ \implies 132=6.28\times R\\ \\ \implies \sf R=\dfrac{132}{6.28}$

$\implies \sf R=21\;cm\;(Approx)$

$\rule{200}{2}$

${\underline{\sf Now,\;we\;will\;find\;area\;of\;ring,}}\\ \\ \implies \sf Area\;of\;ring=Area\;of\;2nd\;circle-Area\;of\;1st\;circle\\ \\ \implies \sf Area\;of\;ring=\pi R^{2}-\pi r^{2}\\ \\ \implies \sf Area\;of\;ring=\pi (R^{2}-r^{2})\\ \\ \implies Area\;of\;ring=3.14(21^{2}-14^{2})\\ \\ \implies \sf Area\;of\;ring=3.14(441-196)\\ \\ \implies \sf Area\;of\;ring=3.14(245)\\ \\ \implies {\boxed{\sf Area\;of\;ring=769.3\;cm^{2}\;(Approx)}}$

Answer 2

Answer:

$\bold\red{Area\:of\:ring}$ = $\bold{770\:{cm}^{2}(Approx.)}$

Step-by-step explanation:

$\bf{Given:-}\begin{cases}\sf{Circumference_1=132\:cm}\\\sf{Circumference_2 = 88\:cm}\\\sf{Area\:of\:ring\:elclosed\:between\:them=?}\end{cases}$

$\bigstar{\boxed{\bold\red{Circumference = 2\pi r}}}$

$\Rightarrow\bf Circumference_1 = 2\pi r_2 \\\\\Rightarrow\bf 132 = 2 \times \dfrac{22}{7}\times r_2 \\\\\Rightarrow\bf 132 = \dfrac{44}{7}\times r_2 \\\\\Rightarrow\bf r_2 = \dfrac{132 \times 7}{44} \\\\\Rightarrow\bf r_2 = \cancel{\dfrac{924}{44}} \\\\\Rightarrow{\boxed{\bf\red{r_2 = 21\:cm\: \: (Approx.)}}}$

$\rule{300}{1}$

$\Rightarrow\bf Circumference_2 = 2\pi r_1 \\\\\Rightarrow\bf 88 = 2\times \dfrac{22}{7}\times r_1 \\\\\Rightarrow\bf r_1 =\dfrac{88 \times 7}{44} \\\\\Rightarrow\bf r_1 =\cancel{\dfrac{616}{44}} \\\\\Rightarrow{\boxed{\bf\red{r_1 = 14\:cm\: \: (Approx.)}}}$

$\rule{300}{1}$

$\bigstar{\boxed{\bf\purple{Area\:of\:circle =\pi{r}^{2} }}}$

$\Rightarrow\bf Area\:of\:ring = Area_2 - Area_1 \\\\\Rightarrow\bf Area\:of\:ring = \pi {(21)}^{2} - \pi{(14)}^{2} \\\\\Rightarrow\bf Area\:of\:ring = (\dfrac{22}{7}\times 441) - (\dfrac{22}{7}\times 196) \\\\\Rightarrow\bf Area\:of\:ring = \dfrac{9702}{7} - \dfrac{4312}{7} \\\\\Rightarrow\bf Area\:of\:ring = \dfrac{9702 - 4312}{7}\\\\\Rightarrow\bf Area\:of\:ring =\cancel{ \dfrac{5390}{7}} \\\\\Rightarrow{\boxed{\bf\red{Area\:of\:ring = 770\:{cm}^{2} \: \: \: (Approx.)}}}$