Find the arc length of a circle with a diameter of 20cm that subtends a central angle of 40. Do not round, leave the answer as an exact value.

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

\[
\text{Arc length} = r \theta
\]

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

### Step 1: Find the radius
The radius \( r \) is half of the diameter. Given the diameter is 20 cm, the radius is:

\[
r = \frac{20}{2} = 10 \, \text{cm}
\]

### Step 2: Convert the central angle to radians
The given angle is \( 40^\circ \). To convert degrees to radians, use the conversion factor:

\[
\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}
\]

Thus:

\[
\theta = 40 \times \frac{\pi}{180} = \frac{40\pi}{180} = \frac{2\pi}{9} \, \text{radians}
\]

### Step 3: Calculate the arc length
Now, substitute \( r = 10 \) cm and \( \theta = \frac{2\pi}{9} \) into the arc length formula:

\[
\text{Arc length} = 10 \times \frac{2\pi}{9} = \frac{20\pi}{9} \, \text{cm}
\]

Thus, the exact arc length is:

\[
\frac{20\pi}{9} \, \text{cm}
\]

Would you like more details on any step?

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Here are 5 related questions:
1. How would the arc length change if the central angle were \(60^\circ\)?
2. What is the circumference of the circle with a diameter of 20 cm?
3. How do you convert radians back to degrees?
4. What is the area of a sector subtended by a \(40^\circ\) angle?
5. How is the arc length affected if the diameter is doubled?

**Tip:** When working with circular measurements, always check whether angles are in degrees or radians!

Learn how to find the exact arc length of a circle with a diameter of 20 cm and a central angle of 40 degrees. This problem involves key geometric concepts such as arc length calculation and radians conversion. Suitable for students in Grades 9-10.

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