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

$r_n=\frac{n^2}{Z} ...\left(1\right)$

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

 $\begin{array}{rcl}{r}_1& =& \frac{1^2}{1}\\ & =& 1 ...\left(2\right)\\ r_3& =& \frac{3^2}{Z}\\ & =& 9 ...\left(3\right)\end{array}$

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

$\begin{array}{rcl}\frac{r_1}{r_3}& =& \frac{1}{9}\\ r_3& =& 9r_1\\ & =& 9\times 5.3\times {10}^{-11} \mathrm{m}\\ & =& 47.7\times {10}^{-11} \mathrm{m}\\ & =& 4.77 \stackrel{\circ }{\mathrm{A}}\end{array}$