Question

The radii of two circles are $8cm$ and $6cm$ respectively. Find the radius of the circle having area equal to the sum of the areas of the two circles

Answer 1

Let $r_1$ and $A_1$ be the radius and area, respectively of first circle and $r_2$ and $A_2$ be the radius and area, respectively of second circle.

Let $r_3$ and $A_3$ be the radius and area, respectively of required circle.

For first circle,

$r_1=8$cm ...Given

$A_1=\pi {r_1}^2$

$\therefore A_1=\pi \times 8^{2}c{m}^2=64\pi c{m}^2$ ...(1)

For second circle,

$r_2=6$cm ...Given

$A_2=\pi {r_2}^2$

$\therefore A_2=\pi \times 6^{2}c{m}^2=36\pi c{m}^2$ ...(2)

Now,for the required circle,

$A_3=A_1+A_2$ ....Given

$\Rightarrow A_3=64\pi +36\pi$ ....From (1) and (2)

$\Rightarrow A_3=100\pi c{m}^2$

$\therefore \pi {r_3}^2=100\pi$

$\Rightarrow r_3^2=100$

$\Rightarrow r_3=10$

The radius of the required circle is $10$ cm.