Question

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

Answer 1

Hint: According to the question given in the question we have to find the radius of the circle when the radii of two circles are 8cm and 6cm respectively and the circle having area equal to the sum of the area of two circles. So, first of all we have to let the radius of the bigger circle have the same variable.
Now, we have to find the area of the circle having radius is 8cm with the help of the formula to find the area of the circle as given below:

Formula used: Area of circle $ = \pi {R^2}$…………………(1)
Where, R is the radius of the circle given.
Now, same as we have to find the area of the circle having radius 6cm 8cm with the help of the formula to find the area of the circle as given above.
Now, as given in the question, the sum of the area of the circles having radius 6cm and 8cm is equal to the area of the bigger circle. So after finding the area we can obtain the value of the radius of the bigger circle.

Complete step-by-step answer:
Given,
Radii of two circles are 8cm and 6cm.
Step 1: First of all we have to let the radius of the bigger circle.
Let the area of the bigger circle$ = xcm$
Step 2: Now, we have to find the area of the circle having radius 8cm with the help of the formula (1) as mentioned in the solution hint. Hence, on substituting the radius in formula (1).
$
   = \pi {(8)^2} \\
   = 64\pi c{m^2}
 $
Step 3: Now, same as the step 1 we have to find the area of the circle having radius 6cm with the help of the formula (1) as mentioned in the solution hint. Hence, on substituting the radius in formula (1).
$
   = \pi {(6)^2} \\
   = 36\pi c{m^2}
 $
Step 4: Now, we have to find the area of the bigger circle of radius r cm with the help of the formula (1) as mentioned in the solution hint.
Area of bigger circle $ = \pi {r^2}$
Step 5: Now, according to the solution mentioned in the solution, the sum of the area of the circles having radius 6cm and 8cm is equal to the area of the bigger circle. Hence,
$ \Rightarrow 64\pi + 36\pi = \pi {r^2}$
On eliminating $\pi $ from both the terms of the expression as obtained just above,
$
   \Rightarrow 100 = {r^2} \\
   \Rightarrow {r^2} = 100 \\
   \Rightarrow r = \sqrt {100} \\
   \Rightarrow r = 10cm
 $

Hence, with the help of the formula (1) to find the area of a circle as mentioned in the solution hint we have obtained the radius of required circle $r = 10cm$

Note: The area is not actually the part of the circle remember the circle is just a locus of points and the area is enclosed inside the locus of points.
To find the area of a circle we have to recall that the relationship between the circumference of a circle and its diameter is always the same ratio.