Arc of a circle is of length 5pie cm and sector it bones as an area of 25 pieCM square it radius is

To find the radius \( r \) of the circle, we will use the formulas for the arc length and the area of a sector.

### Given:
- Arc length \( l = 5\pi \) cm
- Area of the sector \( A = 25\pi \) cm²

### Formulas:
1. **Arc length formula**:
   \[
   l = r \theta
   \]
   where \( r \) is the radius and \( \theta \) is the angle of the sector in radians.

2. **Area of a sector formula**:
   \[
   A = \frac{1}{2} r^2 \theta
   \]
   where \( A \) is the area, \( r \) is the radius, and \( \theta \) is the angle in radians.

### Step 1: Find the angle \( \theta \) using the arc length formula.
From the arc length formula, we can express \( \theta \) as:
\[
\theta = \frac{l}{r} = \frac{5\pi}{r}
\]

### Step 2: Substitute \( \theta \) into the area formula.
Now, substitute \( \theta = \frac{5\pi}{r} \) into the area formula for a sector:
\[
A = \frac{1}{2} r^2 \theta
\]
\[
25\pi = \frac{1}{2} r^2 \cdot \frac{5\pi}{r}
\]
Simplifying the equation:
\[
25\pi = \frac{5\pi r}{2}
\]

### Step 3: Solve for \( r \).
Multiply both sides by 2 to eliminate the fraction:
\[
50\pi = 5\pi r
\]
Divide both sides by \( 5\pi \):
\[
r = 10 \, \text{cm}
\]

### Conclusion:
The radius \( r \) of the circle is **10 cm**.

Would you like further details or have any questions?

### Related Questions:
1. What is the formula for the circumference of a circle?
2. How would you find the angle of the sector in degrees?
3. How do the formulas for arc length and sector area relate to each other?
4. What is the relationship between the radius and the total area of a circle?
5. How would you calculate the diameter of the circle given the radius?

**Tip**: Always express angles in radians when using arc length and sector area formulas.

Arc of a circle is of length 5π cm and sector it bones as an area of 25π cm² it radius is

Learn how to find the radius of a circle given an arc length of 5π cm and a sector area of 25π cm². This problem covers key geometric concepts such as arc length, sector area, and using radian measure. Suitable for students in Grades 9-11.

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