https://file.aiquickdraw.com/mathbot/file/1730195760624wu8lasqy.jpg

To find the **perimeter** and **area** of the shape in the image, we need to observe the following key features:

1. The shape is a **quarter-circle** with radius \( r = 68.5 \, \text{m} \).
2. There is a **90-degree angle** where the quarter-circle meets the straight sides.

### Step 1: Calculate the Perimeter
The perimeter of the quarter-circle consists of:
   - The **arc length** of the quarter-circle.
   - The **two straight sides** that form the radius of the circle.

#### 1. Arc Length of the Quarter-Circle
The formula for the arc length of a quarter-circle is:
\[
\text{Arc length} = \frac{1}{4} \times 2 \pi r = \frac{\pi r}{2}
\]
Substituting \( r = 68.5 \, \text{m} \):
\[
\text{Arc length} = \frac{\pi \times 68.5}{2} \approx 107.64 \, \text{m}
\]

#### 2. Two Straight Sides (Radii)
Since each side is equal to the radius \( r = 68.5 \, \text{m} \), the sum of the two straight sides is:
\[
2 \times 68.5 = 137 \, \text{m}
\]

#### Total Perimeter
Adding the arc length and the two straight sides:
\[
\text{Total Perimeter} = 107.64 + 137 = 244.64 \, \text{m}
\]

### Step 2: Calculate the Area
The area of a quarter-circle is given by:
\[
\text{Area} = \frac{1}{4} \pi r^2
\]
Substituting \( r = 68.5 \, \text{m} \):
\[
\text{Area} = \frac{1}{4} \pi (68.5)^2 \approx \frac{\pi \times 4692.25}{4} \approx 3684.31 \, \text{m}^2
\]

### Final Answer
- **Perimeter:** \( \approx 244.64 \, \text{m} \)
- **Area:** \( \approx 3684.31 \, \text{m}^2 \)

Would you like further details on any specific step or have any questions?

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**Related Questions:**

1. What would the perimeter be if the shape were a half-circle?
2. How would the area change if the radius were doubled?
3. What is the relationship between the radius and the arc length in a quarter-circle?
4. How does the area of a quarter-circle compare to a full circle with the same radius?
5. If the quarter-circle were part of a larger composite shape, how would you calculate the total area?

**Tip:** For circles, remember that arc length and area scale directly with the radius and the square of the radius, respectively.

Learn how to calculate the perimeter and area of a quarter-circle with a radius of 68.5 meters. This guide covers the arc length and area formulas for quarter circles, using step-by-step explanations. Ideal for high school students in Grades 9-11.

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