Question

One rectangle is "framed" within another. Find the area of the shaded region if the "frame" is 3 units wide.
10


the frame is 12 x 10

Answer 1

To find the area of the shaded region (the frame) when one rectangle is framed within another, we can follow these steps:

1. Calculate the area of the outer rectangle.
2. Calculate the area of the inner rectangle.
3. Subtract the area of the inner rectangle from the area of the outer rectangle to find the area of the shaded region.

Given:
- The dimensions of the outer rectangle are 12 units by 10 units.
- The frame around the inner rectangle is 3 units wide.

### Step 1: Area of the Outer Rectangle
The area of the outer rectangle can be calculated using the formula for the area of a rectangle:
\[
\text{Area} = \text{length} \times \text{width}
\]
\[
\text{Area}_{\text{outer}} = 12 \times 10 = 120 \text{ square units}
\]

### Step 2: Dimensions of the Inner Rectangle
Since the frame is 3 units wide all around, the dimensions of the inner rectangle will be:
- Length: \( 12 - 2 \times 3 = 12 - 6 = 6 \) units
- Width: \( 10 - 2 \times 3 = 10 - 6 = 4 \) units

### Step 3: Area of the Inner Rectangle
Now, we calculate the area of the inner rectangle:
\[
\text{Area}_{\text{inner}} = 6 \times 4 = 24 \text{ square units}
\]

### Step 4: Area of the Shaded Region
Finally, we subtract the area of the inner rectangle from the area of the outer rectangle to find the area of the shaded region:
\[
\text{Area}_{\text{shaded}} = \text{Area}_{\text{outer}} - \text{Area}_{\text{inner}} = 120 - 24 = 96 \text{ square units}
\]

The area of the shaded region is \( \boxed{96} \) square units.