Given Radius of first circle = \[{{r}_{1}}\] = \[15\] cm
Given Radius of second circle = \[{{r}_{2}}\] = \[18\] cm
Therefore, Circumference of first circle of radius \[{{r}_{1}}\]= \[2\pi {{r}_{1}}\] = \[30\pi \] cm
Circumference of second circle of radius \[{{r}_{2}}\] = \[2\pi {{r}_{2}}\] = \[36\pi \] cm
Let us assume the radius of the circle = R
From the given question,
Circumference of circle = Circumference of first circle + Circumference of second circle
\[2\pi R\]= \[2\pi {{r}_{1}}\]+ \[2\pi {{r}_{2}}\]
⇒ \[2\pi R\] = \[30\pi \] + \[36\pi \]
⇒ \[66\pi \]
⇒ R = \[33\]
⇒ Radius = \[33\]cm
Therefore ,the required radius of a circle is \[33\] cm.