To solve this problem, we need to calculate the shaded and unshaded areas of the square and find their ratio.

Step 1: Determine the total area of the square.

• The square has a side length of 8 cm (as shown in the diagram).
• The total area of the square is given by: $\text{Total area} = 8 \times 8 = 64 \, \text{cm}^2$

Step 2: Calculate the area of the shaded regions.

The shaded regions are made up of two parts:

1. A smaller square at the top-left corner with side lengths of 4 cm.
2. A rectangular strip on the right-hand side, which is 4 cm by 8 cm.

• Area of the small square: $\text{Area of small square} = 4 \times 4 = 16 \, \text{cm}^2$

• Area of the rectangular strip: $\text{Area of rectangular strip} = 4 \times 8 = 32 \, \text{cm}^2$

• Total shaded area: $\text{Total shaded area} = 16 + 32 = 48 \, \text{cm}^2$

Step 3: Calculate the unshaded area.

• The unshaded area is the remaining part of the square: $\text{Unshaded area} = \text{Total area} - \text{Shaded area} = 64 - 48 = 16 \, \text{cm}^2$

Step 4: Find the ratio of the shaded area to the unshaded area.

• The ratio of the shaded area to the unshaded area is: $\text{Ratio} = \frac{48}{16} = 3$

Thus, the ratio of the shaded area to the unshaded area is 3:1.

Would you like further details or have any questions?

Related Questions:

1. How do you find the area of a rectangle in different units of measurement?
2. What is the formula for the area of any polygon?
3. How can you verify the ratio of two areas mathematically?
4. How does the concept of ratio apply in geometry problems like this?
5. What other methods can you use to solve for shaded vs. unshaded areas in complex shapes?

Tip:

When solving for areas involving composite shapes, break the shape into smaller, manageable parts (e.g., squares and rectangles), calculate their areas individually, and sum them up.