Answer
Circle: $190.8$ inches
Square: $169.2$ inches
Work Step by Step
Let's note:
$r$ = the radius of the circle
$x$ = the length of the square's side.
We are given:
$$\begin{cases}
2\pi r+4x&=360\\
\pi r^2&=x^2.
\end{cases}$$
Use the substitution method to solve the system:
$$\begin{align*}
r&=\dfrac{360-4x}{2\pi}=\dfrac{180-2x}{\pi}\\
\pi \left(\dfrac{180-2x}{\pi}\right)^2&=x^2\\
(180-2x)^2&=\pi x^2\\
180-2x&=\pm\sqrt{\pi}x\\
180&=2x\pm\sqrt{\pi}x\\
180&=x(2\pm\sqrt{\pi})\\
x&=\dfrac{180}{2\pm\sqrt{\pi}}\\
x_1&=\dfrac{180}{2-\sqrt{\pi}}\approx 791\\
x_2&=\dfrac{180}{2+\sqrt{\pi}}\approx 47.7.
\end{align*}$$
Because $x<360$, only the solution $x=47.7$ fits.
Determine the length of the square piece:
$$4x=4(47.7)=190.8.$$
Determine the length of the circle piece:
$$2\pi r=360-190.8.=169.2$$
