Question: I'm struggling with these types of questions lately. Thanks in advance for the help!

Answer: 
The area of a circle with radius $r$ is $A = \pi r^2$, and the circumference is $C = 2\pi r$. Suppose we wish to calculate a sector with central angle $\theta$, where $\theta$ is in radians. Because we are using radians, we know that the arc length cut off by the angle $\theta$ is equal to $r \theta$. Now, consider the proportion $A\_{\text{circle}} : C\_{\text{circle}} :: A\_{\text{sector}} : C\_{\text{sector}}$, where the "circumference" of the sector is the same as the subtended arc length. We can plug into this ratio to find that 
$$\frac{\pi r^2 }{2 \pi r} = \frac{ A\_{\text{sector}}} {r\theta} \implies A\_{\text{sector}} = (1/2)\theta r^2$$ 

This is how we produce the formula given by André. As stated, plug in for everything that you know and try to find the unknown value.
We very well know that the area of a sector having radius $R$ & (aperture) angle $\alpha$ at the center is given as $$\color{blue}{A\_{sector}=\frac{1}{2}\alpha R^2}$$ As per your question, radius $R=13$ inch, Area $A=42.25\pi$ square inches, setting the corresponding values we get $$\frac{1}{2}\alpha (13)^2=42.25\pi$$ $$\alpha=\frac{2\times42.25\pi}{169}$$ $$\alpha=\frac{84.5\pi}{169}=0.5 \pi$$$$=\color{blue}{\frac{\pi}{2}=90^o}$$