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Let's work through the problem shown in the image.

### Given:
- A circle with radius \( r = 8 \) cm.
- A sector of the circle with a central angle of 3 radians.

We need to find:
1. The length of the arc from \( A \) to \( B \) passing through \( C \).
2. The area of the shaded sector.

#### Part (a) - Arc Length

The formula for the length of an arc is:

\[
L = r \theta
\]

Where:
- \( L \) is the arc length,
- \( r \) is the radius,
- \( \theta \) is the angle in radians.

Here, \( r = 8 \) cm and \( \theta = 3 \) radians. Plugging in the values:

\[
L = 8 \times 3 = 24 \text{ cm}
\]

Thus, the length of the arc from \( A \) to \( B \) passing through \( C \) is exactly 24 cm.

#### Part (b) - Area of the Shaded Sector

The formula for the area of a sector is:

\[
A = \frac{1}{2} r^2 \theta
\]

Where:
- \( A \) is the area of the sector,
- \( r \) is the radius,
- \( \theta \) is the angle in radians.

Substituting the known values:

\[
A = \frac{1}{2} \times 8^2 \times 3 = \frac{1}{2} \times 64 \times 3 = 96 \text{ cm}^2
\]

Thus, the area of the shaded sector is exactly \( 96 \, \text{cm}^2 \).

---

Do you want further details or clarifications on this? Here are five related questions to explore:

1. How would the arc length change if the radius was doubled but the angle remained the same?
2. What is the formula for the circumference of a full circle with radius 8 cm?
3. Can you express the area of the entire circle using the radius?
4. How does the formula for sector area change if the angle is measured in degrees?
5. If the angle was given in degrees, how would you convert it to radians?

**Tip:** When working with circles, remember that angles in radians make calculations simpler for arc length and sector area compared to degrees.

A circle of radius 8 cm has a sector whose central angle has radian measure of 3. Find the following exactly: (a) the length of the arc from A to B passing through C. (b) the area of the shaded sector.

Learn how to calculate the exact arc length and area of a sector in a circle with a radius of 8 cm and a central angle of 3 radians. This geometry problem explores the formulas for arc length and sector area, suitable for students in Grades 9-11.

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