Find the area of the shaded region if the "frame" is 2 units wide.


To find the area of the shaded region where the "frame" is 2 units wide, we need to consider the larger rectangle (including the frame) and the smaller rectangle (inside the frame).

Let's denote:
- \( L \): Length of the larger rectangle.
- \( W \): Width of the larger rectangle.
- \( l \): Length of the smaller rectangle.
- \( w \): Width of the smaller rectangle.
- \( f \): Width of the frame.

Given:
- \( f = 2 \) units
- \( l = 6 \) units
- \( w = 2 \) units

The dimensions of the larger rectangle will be the dimensions of the smaller rectangle increased by twice the width of the frame on each side. Thus:

\[ L = l + 2f \]
\[ W = w + 2f \]

Substitute the given values:

\[ L = 6 + 2 \times 2 = 6 + 4 = 10 \, \text{units} \]
\[ W = 2 + 2 \times 2 = 2 + 4 = 6 \, \text{units} \]

Now, we calculate the areas:

\[ \text{Area of the larger rectangle} = L \times W = 10 \times 6 = 60 \, \text{square units} \]
\[ \text{Area of the smaller rectangle} = l \times w = 6 \times 2 = 12 \, \text{square units} \]

Thus, the area of the shaded region is:

\[ \text{Area of the shaded region} = \text{Area of the larger rectangle} - \text{Area of the smaller rectangle} \]
\[ \text{Area of the shaded region} = 60 - 12 = 48 \, \text{square units} \]

Therefore, the area of the shaded region is \( 48 \, \text{square units} \).

Do you want any further details or have any questions?

Here are some related questions:
1. How do you calculate the area of a frame if its width changes?
2. What happens to the area of the shaded region if the frame width is increased?
3. How do you determine the dimensions of the larger rectangle if only the smaller rectangle's dimensions and frame width are given?
4. How can the concept of area help in real-life applications such as tiling or painting?
5. What are other geometric shapes where the area of a shaded region can be calculated similarly?
6. How would you find the area of a shaded region if the inner rectangle is not centered?
7. What is the significance of knowing the area of shaded regions in design and architecture?
8. How do the perimeter and area of a rectangle change when the frame width is increased?

**Tip:** When dealing with composite shapes, break down the problem into smaller parts, calculate each part's area separately, and then combine the results to find the total area.

Learn how to calculate the area of the shaded region when a frame of 2 units width surrounds a smaller rectangle inside a larger rectangle. Detailed step-by-step solution provided.

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