The forces acting on the airplane are gravity ($F_g = mg$) acting vertically downwards, and the aerodynamic lift ($L$) acting at an angle $\theta$ with respect to the vertical.

We can resolve the lift force into its vertical and horizontal components:

$$L_y = L \cos\theta$$ $$L_x = L \sin\theta$$

For the plane to maintain level flight (no vertical acceleration), the vertical component of the lift must balance the force of gravity:

$$\sum F_y = L_y - mg = 0 \implies L \cos\theta = mg$$

The horizontal component of the lift provides the centripetal force required for the airplane to move in a circle of radius $r$ at speed $v$:

$$\sum F_x = L_x = ma_c = \frac{mv^2}{r} \implies L \sin\theta = \frac{mv^2}{r}$$

To find the radius $r$, we can divide the horizontal force equation by the vertical force equation to eliminate the lift force $L$ and mass $m$:

$$\frac{L \sin\theta}{L \cos\theta} = \frac{mv^2/r}{mg}$$ $$\tan\theta = \frac{v^2}{rg}$$

Solving for the radius $r$:

$$r = \frac{v^2}{g \tan\theta}$$