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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.

Work out the perimeter of the given compound shape. Give the answer correct to 2 decimal places. Use π button on your calculator.

Calculate the perimeter of a compound shape that combines a quarter of a circle with a rectangle. Includes detailed steps using the circumference formula for circles and basic perimeter concepts for rectangles. Ideal for Grade 7 students.

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