# The area enclosed between two concentric circles is $770c{m^2}$. If the radius of the outer circle is $21cm$, calculate the radius of the inner circle.

A. 7cm

B. 14cm

C. 2.1cm

D. 35cm

Answer

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367.2k+ views

Hint: Try to find the area enclosed between two concentric circles and then put the value of R.

Let radius of the inner circle be $r$and radius of the outer circle be $R = 21cm$

Area enclosed between the two concentric circles $ = \pi \left( {{R^2} - {r^2}} \right) = 770c{m^2}$

$

\Rightarrow 770 = \pi \left( {{R^2} - {r^2}} \right) \\

\Rightarrow 770 = \dfrac{{22}}{7}\left( {{{21}^2} - {r^2}} \right) \\

\Rightarrow \dfrac{{770 \times 7}}{{22}} = 441 - {r^2} \\

\Rightarrow 245 = 441 - {r^2} \\

\Rightarrow {r^2} = 196 \\

\Rightarrow r = \sqrt {196} \\

\Rightarrow r = 14 \\

$

Thus, the correct option is B.

Note: Two circles are concentric if their centers coincide. The area enclosed between two concentric circles is also referred to as the annulus or circular ring. The area of the annulus equals the area of the larger circle minus the area of the smaller circle.

Let radius of the inner circle be $r$and radius of the outer circle be $R = 21cm$

Area enclosed between the two concentric circles $ = \pi \left( {{R^2} - {r^2}} \right) = 770c{m^2}$

$

\Rightarrow 770 = \pi \left( {{R^2} - {r^2}} \right) \\

\Rightarrow 770 = \dfrac{{22}}{7}\left( {{{21}^2} - {r^2}} \right) \\

\Rightarrow \dfrac{{770 \times 7}}{{22}} = 441 - {r^2} \\

\Rightarrow 245 = 441 - {r^2} \\

\Rightarrow {r^2} = 196 \\

\Rightarrow r = \sqrt {196} \\

\Rightarrow r = 14 \\

$

Thus, the correct option is B.

Note: Two circles are concentric if their centers coincide. The area enclosed between two concentric circles is also referred to as the annulus or circular ring. The area of the annulus equals the area of the larger circle minus the area of the smaller circle.

Last updated date: 01st Oct 2023

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