Find the surface area of each shape (see 3.14)

From the image, we can identify the following dimensions:

• The height of the prism, $h = 6 \text{ cm}$.
• The side length of the hexagonal base, $s = 2 \text{ cm}$.
• The apothem (distance from the center to the midpoint of a side) of the hexagonal base, $a = 3 \text{ cm}$.

The perimeter of the hexagonal base ($P$) is the sum of the lengths of its 6 sides.

$$P = 6 \times s = 6 \times 2 \text{ cm} = 12 \text{ cm}$$

The area of one hexagonal base ($A_{base}$) is given by the formula $A_{base} = \frac{1}{2} \times P \times a$.

$$A_{base} = \frac{1}{2} \times 12 \text{ cm} \times 3 \text{ cm}$$ $$A_{base} = 18 \text{ cm}^2$$

The lateral surface area ($A_{lateral}$) is the area of all the rectangular faces around the prism. It is calculated by multiplying the perimeter of the base by the height of the prism.

$$A_{lateral} = P \times h$$ $$A_{lateral} = 12 \text{ cm} \times 6 \text{ cm}$$ $$A_{lateral} = 72 \text{ cm}^2$$

The total surface area ($SA$) of the prism is the sum of the areas of the two bases and the lateral surface area.

$$SA = 2 \times A_{base} + A_{lateral}$$ $$SA = 2 \times 18 \text{ cm}^2 + 72 \text{ cm}^2$$ $$SA = 36 \text{ cm}^2 + 72 \text{ cm}^2$$ $$SA = 108 \text{ cm}^2$$

The calculation shown in the image, $3.14 (6+12+2) = 48.88 \text{ cm}^2$, does not correspond to the surface area of the hexagonal prism with the given dimensions, as $3.14 \times 20 = 62.8$.

The final answer is $108 \text{ cm}^2$.