https://file.aiquickdraw.com/mathbot/file/1725571635874qeixc030.jpg

To calculate both the length \( s \) of the arc and the area \( A \) of the sector for the given circle, we'll need to extract the following from the image:

1. The radius \( r \) of the circle.
2. The central angle \( \theta \) in radians (or degrees).

### From the image:
- The radius \( r \) is 6 units.
- The central angle \( \theta \) is \( \frac{\pi}{3} \) radians (this corresponds to 60°).

Now, let's compute the two values.

### 1. Arc Length \( s \):
The formula for the arc length of a sector is:
\[
s = r \cdot \theta
\]

### 2. Area of the Sector \( A \):
The formula for the area of a sector is:
\[
A = \frac{1}{2} r^2 \cdot \theta
\]

I'll calculate these values now.The calculations are as follows:

1. The length of the arc \( s \) is approximately \( 6.28 \) units.
2. The area of the sector \( A \) is approximately \( 18.85 \) square units.

Would you like more details or have any questions about these calculations?

Here are five related questions for further exploration:

1. How would the arc length change if the central angle were doubled?
2. What would the area of the full circle be if this sector represents a fraction of it?
3. How do you calculate the perimeter of the sector?
4. Can you find the angle if only the arc length and radius are given?
5. How does changing the radius affect both the arc length and area?

**Tip**: Always ensure that the angle is in radians when using these formulas for arc length and sector area.

Learn how to calculate the arc length and area of a sector for a circle with a 6 unit radius and a central angle of π/3 radians. This problem covers essential geometry concepts, including the formulas for arc length and sector area. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation