1. Find the measure in radians and degrees of the central angle of a circle subtended by the given are. Round approximate answers to the nearest hundredth.

a. r = 2 inches, s = 8 inches

b. r = 35.8 meters, s = 84.3 meters

To solve for the central angle \(\theta\) subtended by an arc, we can use the following formula:

\[
\theta = \frac{s}{r}
\]

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

After finding the angle in radians, we can convert it to degrees using the following formula:

\[
\theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi}
\]

### Part (a): \(r = 2\) inches, \(s = 8\) inches

#### Step 1: Find the angle in radians
\[
\theta = \frac{8}{2} = 4 \text{ radians}
\]

#### Step 2: Convert radians to degrees
\[
\theta_{\text{degrees}} = 4 \times \frac{180}{\pi} \approx 4 \times 57.30 = 229.20^\circ
\]

So, the central angle is approximately **4 radians** or **229.20 degrees**.

---

### Part (b): \(r = 35.8\) meters, \(s = 84.3\) meters

#### Step 1: Find the angle in radians
\[
\theta = \frac{84.3}{35.8} \approx 2.36 \text{ radians}
\]

#### Step 2: Convert radians to degrees
\[
\theta_{\text{degrees}} = 2.36 \times \frac{180}{\pi} \approx 2.36 \times 57.30 = 135.21^\circ
\]

So, the central angle is approximately **2.36 radians** or **135.21 degrees**.

---

### Summary:
- (a) \(4\) radians or \(229.20^\circ\)
- (b) \(2.36\) radians or \(135.21^\circ\)

Would you like a detailed explanation on any step or have further questions?

---

Here are 5 related questions:
1. How do you derive the formula for arc length given the central angle?
2. What is the difference between radian measure and degree measure?
3. Can the formula for central angle be used in ellipses? Why or why not?
4. How would the angle change if the radius was doubled but the arc length stayed the same?
5. What are practical applications of arc length and central angle calculations?

**Tip:** Always use the radian form in trigonometric equations, as radians are the natural angle measure for many mathematical contexts.

1. Find the measure in radians and degrees of the central angle of a circle subtended by the given arc. Round approximate answers to the nearest hundredth.

a. r = 2 inches, s = 8 inches

b. r = 35.8 meters, s = 84.3 meters

Learn how to find the central angle of a circle given the radius and arc length. This guide covers the formula for calculating the angle in both radians and degrees. Examples include r = 2 inches, s = 8 inches and r = 35.8 meters, s = 84.3 meters. Suitable for high school geometry students.

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