The radii of two circles are 19 cm and 9 cm respectively.

The circumference of the circle having radius

$19 \mathrm{~cm}=2 \pi \times 19 \mathrm{~cm}=38 \pi \mathrm{cm}(\because \mathrm{r}=19 \mathrm{~cm})$

The circumference of the circle having radius

$9 \mathrm{~cm}=2 \pi \times 9 \mathrm{~cm}=18 \pi \mathrm{cm} \quad(\because \mathrm{r}=9 \mathrm{~cm})$

Sum of the circumferences of the two circles

$=(38 \pi+18 \pi) \mathrm{cm}=56 \pi \mathrm{cm}$

Therefore, if r cm be the radius of the circle which has circumference equal to the sum of the circumference of the two given circles, then

$2 \pi r=56 \pi \Rightarrow r=28$

Hence, the required radius is 28 cm.