the radius of a circle is 5 and the diameter is 10 and 1/4 of the circle is cut off Find the arc length of the partial circle.

To find the arc length of the partial circle, we'll use the formula for the arc length of a circle:

\[
\text{Arc Length} = \theta \times r
\]

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

### Step 1: Find the central angle \(\theta\)
Since \(\frac{1}{4}\) of the circle is cut off, the central angle \(\theta\) corresponds to a quarter of the full circle. The total angle in a circle is \(2\pi\) radians.

\[
\theta = \frac{2\pi}{4} = \frac{\pi}{2} \text{ radians}
\]

### Step 2: Use the radius \(r = 5\) to find the arc length
Now, plug the values into the formula:

\[
\text{Arc Length} = \frac{\pi}{2} \times 5 = \frac{5\pi}{2} \text{ units}
\]

So, the arc length of the partial circle is \(\frac{5\pi}{2}\) units, which is approximately 7.85 units.

Would you like more details or have any questions? Here are five related questions to consider:

1. How would the arc length change if half the circle were cut off?
2. How do you convert radians to degrees?
3. What is the relationship between the circumference of a circle and its arc length?
4. How would you find the area of the sector corresponding to the \(\frac{1}{4}\) circle cut off?
5. If the radius were doubled, what would the new arc length be for the same central angle?

**Tip:** Remember, the arc length formula \(\theta \times r\) only works when \(\theta\) is in radians. Always convert degrees to radians if necessary.

Learn how to calculate the arc length of a partial circle where the radius is 5 and a quarter of the circle is cut off. Understand the central angle in radians and apply the arc length formula to find the solution. Suitable for high school students studying geometry.

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