Determine the exact measure of central angle in degrees. The central angle that intercepts an arc of length 20 meters on a circle of radius 12 meters.

To determine the exact measure of the central angle in degrees, we can use the relationship between the arc length, the radius of the circle, and the central angle.

The formula for the arc length \( s \) is given by:

\[
s = r \theta
\]

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

Given:
- \( s = 20 \) meters,
- \( r = 12 \) meters.

First, solve for the central angle \( \theta \) in radians:

\[
\theta = \frac{s}{r} = \frac{20}{12} = \frac{5}{3} \text{ radians}
\]

To convert the central angle from radians to degrees, use the conversion factor:

\[
\theta \text{ (in degrees)} = \theta \text{ (in radians)} \times \frac{180^\circ}{\pi}
\]

Thus:

\[
\theta \text{ (in degrees)} = \frac{5}{3} \times \frac{180^\circ}{\pi}
\]

Now, calculate the exact value:

\[
\theta \text{ (in degrees)} = \frac{5 \times 180^\circ}{3 \times \pi} = \frac{900^\circ}{3\pi} = \frac{300^\circ}{\pi}
\]

Therefore, the exact measure of the central angle is:

\[
\boxed{\frac{300^\circ}{\pi}}
\]

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. What is the relationship between the arc length, radius, and central angle in radians?
2. How would the arc length change if the radius was doubled?
3. What is the central angle in degrees for a circle with a radius of 10 meters and an arc length of 25 meters?
4. How can you derive the formula for arc length from the definition of a circle?
5. What is the significance of the central angle in various applications like navigation and astronomy?

**Tip:** Remember that the formula \( s = r \theta \) only works when \( \theta \) is in radians, so always check the units before calculating.

Learn how to determine the exact measure of the central angle in degrees for a circle with a radius of 12 meters and an arc length of 20 meters. This problem involves the arc length formula and radians to degrees conversion, making it suitable for students in Grades 9-11.

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