Certainly! Let's delve into the problem step-by-step.
### Step 1: Understand the Original Shape
You have a cube with each side of length 2 cm. That is important, but our focus here is on hexagon area calculation.
### Step 2: Recognizing the Transformation Factor
The problem states that we need to consider a hexagon and understand its area transformation when enlarged by a scale factor of 3.
### Step 3: Calculate the Area of the Original Hexagon
Firstly, determine the area of a regular hexagon before enlargement:
1. The formula to find the area of a regular hexagon with side length [tex]\( a \)[/tex] is given by:
[tex]\[ A = \frac{3 \sqrt{3}}{2} a^2 \][/tex]
2. Given the original side length is 2 cm:
[tex]\[ a = 2 \][/tex]
3. Substitute [tex]\( a \)[/tex] into the formula:
[tex]\[ A = \frac{3 \sqrt{3}}{2} (2)^2 \][/tex]
[tex]\[ A = \frac{3 \sqrt{3}}{2} \cdot 4 \][/tex]
[tex]\[ A = 6 \sqrt{3} \text{ cm}^2 \][/tex]
### Step 4: Account for the Scale Factor
Next, consider the effect of enlarging this hexagon by a scale factor of 3:
1. When a geometric shape is scaled by a factor [tex]\( k \)[/tex], its area is scaled by [tex]\( k^2 \)[/tex].
2. Given the scale factor [tex]\( k = 3 \)[/tex]:
The area of the hexagon after scaling will be:
[tex]\[ \text{New Area} = (k^2) \times \text{Original Area} \][/tex]
[tex]\[ \text{New Area} = (3^2) \times (6 \sqrt{3}) \][/tex]
[tex]\[ \text{New Area} = 9 \times (6 \sqrt{3}) \][/tex]
[tex]\[ \text{New Area} = 54 \sqrt{3} \][/tex]
### Step 5: Calculate the Numerical Value for the Enlarged Hexagon's Area
To find the numerical value:
[tex]\[ 54 \sqrt{3} \approx 54 \times 1.732 \][/tex]
[tex]\[ 54 \sqrt{3} \approx 93.528 \text{ cm}^2 \][/tex]
Thus, the area of the enlarged hexagon with a scale factor of 3 is approximately:
[tex]\[ 93.5307 \text{ cm}^2 \][/tex]
### Conclusion
Therefore, if a hexagon with sides of length 2 cm is enlarged with a scale factor of 3, its area will be approximately:
[tex]\[ 93.5307 \text{ cm}^2 \][/tex]
