https://file.aiquickdraw.com/mathbot/file/1727106529701optvtgbb.jpg

The problem is asking for the perimeter of the shaded part of the design, which is composed of two circular sectors with a common center \( O \). The provided dimensions are:

- Outer radius: \( 8 \, \text{cm} \)
- Inner radius: \( 4.5 \, \text{cm} \)
- Central angle: \( 50^\circ \)

### Perimeter of the Shaded Area

To calculate the perimeter, we need to account for:

1. **Arc length of the outer circle**:  
   The arc length \( L_1 \) of the outer sector is given by:
   \[
   L_1 = \theta \times \frac{2\pi r_1}{360}
   \]
   where \( \theta = 50^\circ \) and \( r_1 = 8 \, \text{cm} \).

2. **Arc length of the inner circle**:  
   The arc length \( L_2 \) of the inner sector is:
   \[
   L_2 = \theta \times \frac{2\pi r_2}{360}
   \]
   where \( r_2 = 4.5 \, \text{cm} \).

3. **Two radii**: The straight-line segments connecting the center \( O \) to the two endpoints of the arcs, which are simply the radii \( r_1 \) and \( r_2 \).

Thus, the total perimeter \( P \) is:
\[
P = L_1 + L_2 + r_1 + r_2
\]

Let me calculate this.The perimeter of the shaded part is approximately \( 23.4 \, \text{cm} \) when rounded to three significant figures.

Would you like more details on any of these calculations or have any further questions?

Here are some related questions you might find interesting:

1. How do we derive the formula for the arc length of a circle?
2. What would happen to the perimeter if the central angle was doubled?
3. How would the calculation change if the radii were given in different units, like meters?
4. What is the relationship between the central angle and the arc length?
5. Can we calculate the area of the shaded region instead of the perimeter?

**Tip:** Always remember to convert the central angle to radians when dealing with arc length in terms of radians instead of degrees!

Calculate the perimeter of the shaded part formed from two sectors of circles, center O, with outer radius 8 cm, inner radius 4.5 cm, and central angle of 50 degrees. Give your answer correct to 3 significant figures.

Learn how to calculate the perimeter of a shaded part formed from two circular sectors with radii 8 cm and 4.5 cm and a central angle of 50 degrees. This solution covers geometric principles, the arc length formula, and perimeter calculations. Suitable for Grades 8-10.

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