To calculate the arc length of a circle, we use the formula:

$\text{Arc Length} = \theta \cdot r$

where:

• $\theta$ is the central angle in radians,
• $r$ is the radius of the circle.

Step 1: Convert angle to radians

The given angle is $90^\circ$. To convert degrees to radians, use the formula:

$\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$

$\theta = 90^\circ \times \frac{\pi}{180} = \frac{\pi}{2} \, \text{radians}$

Step 2: Substitute into the arc length formula

Now substitute $\theta = \frac{\pi}{2}$ and $r = 20 \, \text{cm}$ into the formula:

$\text{Arc Length} = \theta \cdot r = \frac{\pi}{2} \cdot 20 = 10\pi \, \text{cm}$

Final Answer:

$\text{Arc Length} = 10\pi \, \text{cm} \, (\approx 31.42 \, \text{cm})$

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Would you like a detailed explanation or clarification on this? Here are some follow-up questions:

1. How do we convert radians back to degrees?
2. What is the relationship between the circumference and the arc length?
3. How would the arc length change if the radius were doubled?
4. How can we find the area of the sector subtended by the same angle?
5. Can this formula be extended to ellipses? If so, how?

Tip: Always ensure angles are in radians when using formulas involving trigonometric functions or arc length.