The diameter of a circle is 8 meters. A central angle of the circle intercepts an arc of 12 meters. What is the radian measure of the central angle?

To find the radian measure of the central angle, we use the formula for the arc length of a circle:

\[
s = r \theta
\]

Where:
- \( s \) is the arc length,
- \( r \) is the radius of the circle,
- \( \theta \) is the central angle in radians.

### Step 1: Find the radius
The diameter of the circle is given as 8 meters. The radius \( r \) is half of the diameter:

\[
r = \frac{8}{2} = 4 \, \text{meters}
\]

### Step 2: Use the arc length formula
We are given that the arc length \( s \) is 12 meters. Using the formula \( s = r \theta \), we can solve for \( \theta \):

\[
\theta = \frac{s}{r} = \frac{12}{4} = 3 \, \text{radians}
\]

Thus, the radian measure of the central angle is \( \theta = 3 \, \text{radians} \).

Would you like a more detailed breakdown or have any questions?

Here are 5 related questions:
1. How would you convert the radian measure into degrees?
2. What is the formula for the circumference of a circle, and how would you apply it here?
3. If the arc length were 6 meters, what would the central angle be in radians?
4. What is the area of the circle with a diameter of 8 meters?
5. How can we derive the radian measure formula from the definition of a circle?

Tip: Radians are a dimensionless measure since they are the ratio of two lengths (arc length and radius).

Learn how to find the radian measure of a central angle when given the diameter of 8 meters and an arc length of 12 meters. This solution involves applying the arc length formula, key concepts of geometry, and understanding radians. Ideal for high school students (Grades 9-12).

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