816 Polar moment of inertia and radius of gyration at one corner of rectangle

Moment of inertia about the x-axis
$I_x = \dfrac{bh^3}{3} = \dfrac{6(3^3)}{3}$

$I_x = 54 \, \text{ in.}^4$
 

Moment of inertia about the y-axis
$I_y = \dfrac{hb^3}{3} = \dfrac{3(6^3)}{3}$

$I_y = 216 \, \text{ in.}^4$
 

Polar moment of inertia
$J = I_x + I_y = 54 + 216$

$J = 270 \, \text{ in.}^4$           answer
 

Radius of gyration about the corner
$k_z = \sqrt{\dfrac{J}{A}} = \sqrt{\dfrac{270}{3(6)}}$

$k_z = 3.873 \, \text{ in.}$           answer