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We are tasked with finding the angle \(\theta\) of the sector in radians. The sector forms the curved surface of the cone, with the given dimensions:

- Radius of the base of the cone \( r = 5 \, \text{cm} \)
- Height of the cone \( h = 8 \, \text{cm} \)
- Slant height \( l \) of the cone, which we can calculate using the Pythagorean theorem:
  \[
  l = \sqrt{r^2 + h^2} = \sqrt{5^2 + 8^2} = \sqrt{25 + 64} = \sqrt{89} \approx 9.43 \, \text{cm}
  \]

### Step 1: Relating the sector's radius to the cone's slant height
The radius of the sector is equal to the slant height of the cone, so:
\[
\text{Radius of sector} = l = 9.43 \, \text{cm}
\]

### Step 2: Formula for the arc length
The arc length of the sector is equal to the circumference of the base of the cone. The formula for the arc length is:
\[
\text{Arc length} = r \times \theta
\]
where \( r = 5 \, \text{cm} \) is the radius of the cone's base, and \(\theta\) is the angle of the sector in radians.

The circumference of the cone's base is:
\[
\text{Circumference} = 2\pi \times r = 2\pi \times 5 = 10\pi \, \text{cm}
\]

Thus, we can set the arc length equal to the base's circumference:
\[
l \times \theta = 10\pi
\]
Substitute \( l = 9.43 \):
\[
9.43 \times \theta = 10\pi
\]

### Step 3: Solve for \(\theta\)
\[
\theta = \frac{10\pi}{9.43} \approx 3.33 \, \text{radians}
\]

Therefore, the angle \(\theta\) of the sector is approximately \( 3.3 \, \text{radians} \) when rounded to one decimal place.

Would you like any further details or clarifications? 

Here are some additional related questions:
1. How does the slant height of a cone relate to its base and height?
2. What is the general formula for the angle of a sector when forming a cone?
3. How do you convert degrees to radians, and why is it important in geometry?
4. Can you derive the formula for the volume of a cone using its base radius and height?
5. What are some real-life applications of cones in mathematics?

**Tip:** Always check if the problem involves radians or degrees to avoid confusion in angle measurements.

A cone of base radius 5 cm and height 8 cm is formed from a sector. Find the angle θ of the sector in radians.

Learn how to find the angle θ of a sector in radians that forms the curved surface of a cone with base radius 5 cm and height 8 cm. Step-by-step guide using the Pythagorean theorem and arc length formula. Suitable for students in Grades 9-12.

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