the angle lies at the center of a circle and subtends an arc of the circle. Find the missing angle measure, circle radius, or arc length.
1.) Angle 5pi/8 radius 2 arc length ?
2.) Angle ? radius 14 arc length 7

Let's solve each problem step by step using the relationship between the angle \(\theta\) in radians, the radius \(r\), and the arc length \(s\), which is given by the formula:

\[
s = r \times \theta
\]

### 1) Angle \( \frac{5\pi}{8} \), Radius \( r = 2 \), Arc Length \( s = ? \)

Using the formula:

\[
s = r \times \theta = 2 \times \frac{5\pi}{8} = \frac{10\pi}{8} = \frac{5\pi}{4}
\]

So, the arc length \( s = \frac{5\pi}{4} \) units.

### 2) Angle \( \theta = ? \), Radius \( r = 14 \), Arc Length \( s = 7 \)

Using the formula, solve for \( \theta \):

\[
s = r \times \theta \quad \Rightarrow \quad 7 = 14 \times \theta
\]

\[
\theta = \frac{7}{14} = \frac{1}{2} \text{ radians}
\]

So, the angle \( \theta = \frac{1}{2} \) radians.

### Summary:
1. Arc length for the first problem: \( \frac{5\pi}{4} \) units.
2. Angle for the second problem: \( \frac{1}{2} \) radians.

Would you like to go over any specific details or ask further questions?

Here are 5 related questions to deepen your understanding:
1. How would the arc length change if the radius were doubled in the first problem?
2. What would be the arc length in the second problem if the angle were \( \pi/3 \) radians?
3. How can the concept of radians help in understanding angular velocity?
4. What is the relationship between the degree measure of an angle and its radian measure?
5. How would the arc length change in the first problem if the angle were \( \pi/4 \)?

**Tip:** Always remember that \( \theta \) in radians is the ratio of the arc length to the radius, making it a dimensionless quantity, which is useful in many applications of trigonometry and physics.

The angle lies at the center of a circle and subtends an arc of the circle. Find the missing angle measure, circle radius, or arc length. 1.) Angle 5pi/8 radius 2 arc length ? 2.) Angle ? radius 14 arc length 7

Learn how to calculate the arc length and angle measure in a circle using the relationship between the radius and central angle. This detailed explanation covers the arc length formula and the central angle theorem, suitable for high school students in Grades 9-12.

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