Question

the radii of two circles are 48 cm and 13 cm find the area of the circle which has its circumference equal to the difference of the circumferences of the given two circles

Answer 1

Radius of first circle = 48 cm
Circumference of first circle = 2*pi*48
= 96 pi
Radius of second circle = 13 cm
Circumference of second circle = 2*pi*13
= 26 pi

Circumference of third circle = Circumference of first circle - Circumference of second circle

Circumference of third circle = 96pi - 26pi
2*pi *R = 80 pi
=> R= 40 cm


Area of third circle = pi *r^2
= 3.14 * 40 * 40
= 314 * 16
= 5024 cm^2

Answer 2

Given:−

• Radii of two circles = 48cm and 13cm

$\underline{\sf \ \ \ \star\ To\ Find :- \ \ \ \ \ \ \ }$

• We have to find out the Area of that circle whose circumference is equal to the Difference of the circumference of given two circles

$\underline{\sf \ \ \ \star\ Solution :- \ \ \ \ \ \ \ }$

Find the Circumference of the two circles

$\underline{\boxed{\sf{\dag\ \ Circumference\ of \ circle = 2 \pi r}}}$

Find the circumference of circle whose radius is 48cm

$\begin{gathered}\dashrightarrow\sf Circumference\ of \ Circle_1= 2\times \dfrac{22}{7}\times 48\\ \\ \\ \dashrightarrow\sf Circumference \ of \ Circle_1={\underline{\boxed{\purple{\sf \dfrac{2112}{7}}}}}\end{gathered}$

Find the circumference of circle whose radius is 13cm

$\begin{gathered}\dashrightarrow\sf Circumference\ of \ Circle_2= 2\times \dfrac{22}{7}\times 13\\ \\ \\ \dashrightarrow\sf Circumference \ of \ Circle_2={\underline{\boxed{\red{\sf \dfrac{572}{7}}}}}\end{gathered}$

Now find out the circumference of new circle which is equal to the Difference of $\sf C_1 - C_2C$

$\underline{\sf{\maltese \ Circumference\ of \ new \ circle= C_1- C_2}}$

$\begin{gathered}:\implies\sf Circumference\ of \ new \ circle = \bigg[ \dfrac{2112}{7}\bigg]- \bigg[\dfrac{572}{7}\bigg]\\ \\ \\ :\implies\sf C.\ of \ new \ circle = \cancel{\dfrac{1540}{7}}\\ \\ \\ :\implies\sf C.\ of \ new \ circle = {\underline{\boxed{\purple{\sf 220cm}}}}\end{gathered}$

Now we have to find the Area of new circle

Find out the radius !

$\begin{gathered}\dashrightarrow\sf Circumference\ of \ circle= 2 \pi r\\ \\ \\ \dashrightarrow\sf 220= 2\times \dfrac{22}{7}\times r \\ \\ \\\dashrightarrow\sf r= \dfrac{\cancel{220}\times 7}{\cancel{44}}\\ \\ \\ \dashrightarrow\sf r= 5\times 7\\ \\ \\\dashrightarrow{\underline{\boxed{\sf{\blue{ radius= 35cm}}}}}\end{gathered}$

Now find the Area of new circle

$\underline{\boxed{\sf{\ Area\ of \ circle= \pi r^2 }}}$

$\begin{gathered}\dashrightarrow\sf Area \ of \ circle= \dfrac{22}{\cancel{7}}\times \cancel{35}\times 35\\ \\ \\ \dashrightarrow\sf Area\ of \ circle = 22\times 5\times 35\\ \\ \\ \dashrightarrow\sf Area_{circle}= {\underline{\boxed{\sf{\purple{3850 cm^2}}}}}\end{gathered}$

$\underline{\underline{\textsf{ Area \ of \ new \ circle = {\textbf{3850sq.cm}}}}}$

$\underline{\sf{\bigstar\ Alternate\ Method \ To \ find \ Radius \ of \ new \ circle }}$

$\begin{gathered}\dashrightarrow\sf C_1- C_2= C_{new}\\ \\ \\ \dashrightarrow\sf 2\pi r_1-2\pi r_2= 2\pi r\\ \\ \\ \dashrightarrow\sf 2\pi(r_1-r_2)= 2\pi r\\ \\ \\ \dashrightarrow\sf \cancel{2 \pi}(48-13)= \cancel{2 \pi } r\ \ \ \ \Big[\therefore\ r_1=48\ ; \ r_2= 13 \Big]\\ \\ \\ \dashrightarrow{\boxed{\sf 35= r}}\end{gathered}$

★By using this You can easily find the area of the new circle !