Perimeter of rectangular metal sheet

Total area of sheet $A = 135$ sq in. Given: Unpainted Area $= \frac{2}{3} \times \text{Painted Area}$. Total Area $= \text{Painted Area} + \text{Unpainted Area} = \frac{5}{3} \times \text{Painted Area} = 135$. $\text{Painted Area (Circle)} = 135 \times \frac{3}{5} = 81\text{ sq in}$

Since circle touches two opposite sides of the rectangle, width $w = d = 2r$. $\text{Area of circle} = \pi r^2 = 81 \implies r = \frac{9}{\sqrt{\pi}} \implies w = \frac{18}{\sqrt{\pi}}$

Since total area $l \times w = 135$: $l = \frac{135}{18/\sqrt{\pi}} = \frac{15\sqrt{\pi}}{2}$

Perimeter of rectangle $= 2(l + w) = 2\left(\frac{15\sqrt{\pi}}{2} + \frac{18}{\sqrt{\pi}}\right) = 15\sqrt{\pi} + \frac{36}{\sqrt{\pi}} = 3\sqrt{\pi}\left(5 + \frac{12}{\pi}\right)$.