Let's analyze the problem:

1. Square: The square has an area of $36 \, \text{cm}^2$. This tells us that the side length of the square is:

$\text{Side length} = \sqrt{36} = 6 \, \text{cm}.$

2. Triangle: One of the components in the figure is a triangle formed by one of the diagonals of the square. The triangle is a right-angled triangle with both legs measuring 6 cm each.

3. Quadrant: The curved part is a quarter of a circle (quadrant). Its radius is equal to the side of the square, which is 6 cm. The circumference of a full circle with radius 6 cm is:

$\text{Circumference} = 2 \pi \times 6 = 12\pi \, \text{cm}.$

Since the quadrant is one-fourth of a full circle, its arc length is:

$\text{Arc length} = \frac{1}{4} \times 12\pi = 3\pi \, \text{cm}.$

Finding the Perimeter of the Shaded Region

The shaded region consists of:

• The arc length of the quadrant: $3\pi \, \text{cm}$.
• The two straight segments of the triangle (the vertical and horizontal edges), each measuring 6 cm.

Thus, the total perimeter of the shaded region is:

$\text{Perimeter} = 3\pi + 6 + 6 = 3\pi + 12 \, \text{cm}.$

Final Answer

The perimeter of the shaded part of the figure is:

$\boxed{3\pi + 12 \, \text{cm}}$

Would you like further details or have any questions?

Related Questions:

1. What is the area of the shaded region in the figure?
2. How do you find the diagonal length of the square?
3. If the radius of the quadrant were doubled, what would be its new arc length?
4. How would the perimeter change if only half the circle was considered?
5. Can you calculate the perimeter for a square of area 49 cm²?

Tip:

For problems involving circles and parts of circles, remember that the arc length formula is $\frac{\theta}{360^\circ} \times 2\pi r$, where $\theta$ is the central angle in degrees.