A cube has all side 2 cm. What will the area of the hexagon be,if it is enlarge with scale factor 3?

First, we need to find the area of the hexagon when it is enlarged with a scale factor of 3.

The area of a hexagon can be calculated using the formula:

Area = (3√3/2) * s^2

Where s is the length of one side of the hexagon.

Since the cube has all sides of 2 cm, the length of one side of the hexagon will be equal to the diagonal of one face of the cube. Using the Pythagorean theorem, we can calculate the diagonal of the face of the cube:

d = √(2^2 + 2^2)
d = √(4 + 4)
d = √8
d = 2√2

Now, we can calculate the area of the hexagon when it is enlarged with a scale factor of 3:

s = 2√2 * 3
s = 6√2

Area = (3√3/2) * (6√2)^2
Area = (3√3/2) * 36 * 2
Area = (3√3/2) * 72
Area = 108√3 cm^2

Therefore, the area of the hexagon when it is enlarged with a scale factor of 3 will be 108√3 cm^2.

Step 1: Find the surface area of the original cube.

Surface area of a cube = 6 - (side length)^2
Surface area = 6 - (2 cm)^2
Surface area = 6 - 4 cm^2
Surface area = 24 cm^2

Step 2: Find the scale factor for the surface area.

Scale factor = (New surface area) / (Original surface area)
Scale factor = (3)^2
Scale factor = 9

Step 3: Find the new surface area of the enlarged cube.

New surface area = Original surface area - Scale factor
New surface area = 24 cm^2 - 9
New surface area = 216 cm^2

Step 4: Find the area of one face of the enlarged cube.

Area of one face = New surface area / 6
Area of one face = 216 cm^2 / 6
Area of one face = 36 cm^2

Step 5: The enlarged cube has 6 faces, so the total area of the hexagon will be:

Area of hexagon = 6 - Area of one face
Area of hexagon = 6 - 36 cm^2
Area of hexagon = 216 cm^2

Therefore, the area of the hexagon after enlarging the cube with a scale factor of 3 will be 216 cm^2.