To solve this problem, we need to determine the length $EG$, which is given as $6 \pi$ inches. The diagram shows a circle with center $O$, radius $OE = OF$, and $\angle EOF = 30^\circ$. Since $EG$ is an arc spanning this angle, we'll interpret $6 \pi$ inches as the arc length of $EG$ over $30^\circ$.

Step-by-Step Solution:

1. Identify the Arc Length Formula: The arc length $s$ of a circle is given by: $s = r \theta$ where $r$ is the radius and $\theta$ is the angle in radians.

2. Convert Degrees to Radians: Since $\angle EOF = 30^\circ$, we convert this to radians: $\theta = 30^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{6}$

3. Set Up the Equation: We know that $s = 6 \pi$ and $\theta = \frac{\pi}{6}$. Plugging these into the arc length formula: $6 \pi = r \cdot \frac{\pi}{6}$

4. Solve for the Radius $r$: Divide both sides by $\frac{\pi}{6}$ to isolate $r$: $r = \frac{6 \pi}{\frac{\pi}{6}} = 6 \times 6 = 36 \text{ inches}$

5. Confirm the Answer: The radius $OE$ (or $OF$) of the circle is $36$ inches.

Final Answer:

The radius $OE = OF = 36$ inches.

Would you like more details on any of these steps or have any further questions?

Here are some related questions to explore:

1. What is the formula for converting degrees to radians?
2. How do you derive the arc length formula $s = r \theta$?
3. If the angle $\angle EOF$ were $60^\circ$, what would the arc length $EG$ be?
4. How would the solution change if $EG$ spanned $90^\circ$ instead of $30^\circ$?
5. Can you explain why dividing by $\frac{\pi}{6}$ in step 4 is the correct approach?

Tip: Remember that angles in radians make many formulas in trigonometry and geometry simpler to apply, especially for arc lengths and sector areas.