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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.

If EG has a length of 6π inches, find the radius of the circle.

Learn how to find the radius of a circle when given the arc length (6π inches) and a central angle of 30°. This problem explores geometry concepts, including the arc length formula and degree-to-radian conversion, with a detailed solution for high school students.

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