The radius of inner most orbit of hydrogen atom is 5 . 3 × 10 ^ -...

The formula to calculate the radius of the ${n}^{th}$ orbit is given by

${r}_{n} = \dfrac{{n}^{2}}{Z} . . . \left(1\right)$

Substitute proper values of the orbit numbers in order to calculate the formula for the radius of the innermost and third allowed orbital radii.

$\begin{cases}{r}_{1} & = & \dfrac{{1}^{2}}{1} \\ = & 1 . . . \left(2\right) \\ {r}_{3} & = & \dfrac{{3}^{2}}{Z} \\ = & 9 . . . \left(3\right)\end{cases}$

Divide equation (2) by equation (3) and solve to calculate the radius of the third allowed orbit.

$\begin{cases}\dfrac{{r}_{1}}{{r}_{3}} & = & \dfrac{1}{9} \\ {r}_{3} & = & 9{r}_{1} \\ & = & 9 \times 5 . 3 \times {10}^{- 11}m \\ & = & 47 . 7 \times {10}^{- 11}m \\ & = & 4 . 77\overset{∘}{A}\end{cases}$