Question

The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has circumference equal to the sum of the circumferences of the two circles.

Answer 1

Hint: Here we have to use the simple formula of the circumference of the circle to find the radius of the resultant circle. Firstly, we will find out the circumference of the two given circles. Then we have to add the two circumferences and equate the resulted circumference to find the radius of the resultant circle.

Formula used:
We will use the formula of the circumference of the circle\[ = 2\pi r\].

Complete step-by-step answer:
It is given that the radius of the two circles is 19 cm and 9 cm respectively.

Now we have to find out the circumference of the circles. Therefore, we get
Circumference of the circle with radius 19 cm \[ = 2\pi {r_1}\]
Substituting \[{r_1} = 19{\rm{cm}}\] in above equation, we get
\[ \Rightarrow \] Circumference of the circle with radius 19 cm \[ = 2\pi \times 19 = 38\pi {\rm{cm}}\]
Circumference of the circle with radius 9 cm \[ = 2\pi {r_2}\]
Substituting \[{r_2} = 9{\rm{cm}}\] in above equation, we get
\[ \Rightarrow \] Circumference of the circle with radius 9 cm \[ = 2\pi \times 9 = 18\pi {\rm{ cm}}\]

Now we have to find out the radius of the circle that has circumference equal to the sum of the circumferences of the two circles. Therefore, we get
Circumference of the resultant circle \[ = 38\pi + 18\pi \]
Substituting the value of circumference of resultant circle, we get
\[\begin{array}{l} \Rightarrow 2\pi {r_0} = 38\pi + 18\pi \\ \Rightarrow 2\pi {r_0} = 56\pi \end{array}\]
Cancelling \[\pi \] from both the side, we get
\[ \Rightarrow 2{r_0} = 56\]
Dividing both side by 2, we get
\[ \Rightarrow {r_0} = 28cm\]
Hence, 28cm is the radius of the resultant circle.

Note: The circumference is a similar concept as the perimeter of the shapes and both are generally measured in a single unit i.e. centimeter or meter. The area is the amount of surface covered by a shape in two dimensions. The area is generally measured in square units.
Area of the circle\[ = \dfrac{{\pi {d^2}}}{4}\], where d is the diameter of the circle. The diameter of a circle is equal to two times the radius of the circle.