Let's break down this problem. The figure consists of part of a circle and part of a square, and we are tasked with finding the perimeter of the figure.

Steps to find the perimeter:

1. Side lengths of the square:

   • The square has a side of 5 units, as given in the image.

2. Quarter-circle:

   • The figure includes a quarter-circle attached to the square. The radius of this quarter-circle is also 5 units (since it's attached to the side of the square).

3. Perimeter Calculation:

   • For the square portion, we include two sides: one vertical side (length 10) and the horizontal side of the square (5 units).
   • The quarter-circle adds the arc length to the perimeter.

4. Arc length of quarter-circle: The arc length of a circle is given by the formula: $\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r$ Since this is a quarter-circle, $\theta = 90^\circ$, and the radius $r = 5$: $\text{Arc Length} = \frac{90}{360} \times 2\pi \times 5 = \frac{1}{4} \times 10\pi = 2.5\pi$ Approximate value: $2.5\pi \approx 7.85$ units.

Total Perimeter:

• Vertical side = 10 units
• Horizontal side of the square = 5 units
• Quarter-circle arc length = $7.85$ units

Adding them up: $\text{Total Perimeter} = 10 + 5 + 7.85 = 22.85 \text{ units}$

So, the perimeter of the figure is approximately 22.85 units.

Do you have any questions or need further details on this solution?

Here are 5 related questions:

1. How would the area of this figure be calculated?
2. What is the perimeter if the radius of the quarter-circle was doubled?
3. How would the perimeter change if the figure was a semicircle instead of a quarter-circle?
4. Can you derive the general formula for the perimeter of a figure made of a square and part of a circle?
5. What would the total perimeter be if the square side length was different from the circle’s radius?

Tip: Always ensure you correctly identify which portions of a figure contribute to the perimeter versus the area to avoid confusion.