The question asks to work out the perimeter of a compound shape, which consists of a quarter of a circle attached to a rectangle. Here's how you can calculate the perimeter:

Step 1: Identify the dimensions.

• The radius of the quarter-circle is $4.5$ meters (since the length of one side of the square is the radius).
• The rectangle has a length of $4.5$ meters and a width of $4.5$ meters.

Step 2: Perimeter of the quarter-circle.

The perimeter of the quarter-circle consists of:

• A curved part (which is a quarter of the circumference of a full circle).
• Two straight sides (the radius of the circle).

Curved part of the quarter-circle:

The formula for the circumference of a full circle is: $C = 2\pi r$ For a quarter of a circle: $\text{Quarter of circumference} = \frac{1}{4} \times 2\pi r = \frac{\pi r}{2}$ Substitute $r = 4.5$: $\text{Curved part} = \frac{\pi \times 4.5}{2} = 7.0686 \, \text{meters} \, (\text{rounded to 4 decimal places}).$

Straight sides of the quarter-circle:

Each of these sides is equal to the radius, which is $4.5$ meters.

Step 3: Add the perimeter of the straight sides of the rectangle.

The other side of the rectangle is also $4.5$ meters (opposite to the quarter-circle curve).

Step 4: Total Perimeter.

Now, sum all the parts: $\text{Perimeter} = \text{Curved part} + 2 \times \text{radius} + \text{other rectangle side}$ $\text{Perimeter} = 7.0686 + 4.5 + 4.5 + 4.5$ $\text{Perimeter} = 20.5686 \, \text{meters}$

Final Answer (to 2 decimal places):

$\boxed{20.57} \, \text{meters}.$

Would you like more details or clarification on any step? Here are a few follow-up questions you can explore:

1. How would you calculate the area of this compound shape?
2. What would the perimeter be if the radius were doubled?
3. How does the perimeter of the quarter-circle compare to the full circle?
4. How would the formula change if this were a semicircle instead of a quarter-circle?
5. Can you derive the formula for the perimeter of a sector of a circle?

Tip:

When working with sectors of circles, always be mindful of the angle subtended by the arc—this determines what fraction of the circumference is involved.