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

Step $1:$ Find the sum of the area of two given circles

It is given that$,$ the radii of the two circles are $8cm$ and $6cm,$

We have to find out the radius of the circle having an area equal to the sum of the areas of the two circles.

Radius $\left(r_1\right)=8cm$

Radius $\left(r_2\right)=6cm$

Area of circle of radius $r$ is $=\pi r^2$

Area of first circle $=\pi {\left(8\right)}^2$

$=64\pi c{m}^2$

Area of second circle $=\pi {\left(6\right)}^2$

$=36\pi c{m}^2$

Sum of the area of two circle$=64\pi +36\pi$

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

Step 2: Find the radius of the third circle

Now, assume that the radius of third circle be $R$

Then area of the third circle $=\pi R^2$

According to the questions

Area of third circle $=$Sum of the area of two circles

$\therefore \pi R^2=100\pi$

$$\begin{gathered} \Rightarrow R^2=100 \\ \Rightarrow R=10cm \end{gathered}$$

Hence, the radius of the circle having an area equal to the sum of the areas of the two circles is $10cm.$