We are given an arc length $s = \frac{25\pi}{8}$ and a central angle $\theta = \frac{5\pi}{4}$, and we need to find the radius $r$ of the circle.

The formula for the arc length $s$ of a circle is:

$s = r \theta$

Where:

• $s$ is the arc length,
• $r$ is the radius, and
• $\theta$ is the angle in radians.

Step-by-step solution:

1. Substitute the known values into the formula:

$\frac{25\pi}{8} = r \cdot \frac{5\pi}{4}$

2. Solve for $r$:
   • First, divide both sides by $\frac{5\pi}{4}$:

$r = \frac{\frac{25\pi}{8}}{\frac{5\pi}{4}}$

3. Simplify the right-hand side:
   • Simplifying the division:

$r = \frac{25\pi}{8} \times \frac{4}{5\pi}$

4. Cancel out $\pi$ from the numerator and denominator:

$r = \frac{25}{8} \times \frac{4}{5} = \frac{25 \times 4}{8 \times 5} = \frac{100}{40} = 2.5$

Final answer:

The exact length of the radius is $r = 2.5$ units.

Would you like further details or explanations?

Here are five related questions you might find interesting:

1. What is the formula for the circumference of a circle?
2. How would the result change if the angle were given in degrees instead of radians?
3. What is the relationship between the radius and the diameter of a circle?
4. How do you find the area of a sector given the arc length and the radius?
5. How do you convert an angle from radians to degrees?

Tip: Always ensure angles are in radians when using formulas involving arc length in circular calculations.