To find the surface area of the steamer basket, which can be visualized as a cylinder with two circular bases and a rectangular lateral surface, we can use the following formula for the surface area of a cylinder:
\[ \text{Surface Area} = 2 \cdot \text{Area of Base} + \text{Lateral Surface Area} \]
Where:
- The area of each circular base is given as 452.16 cm², and since there are two bases, we have \(2 \cdot 452.16\).
- The area of the rectangular lateral face (which is the lateral surface area) is given as 678.24 cm².
Now we can calculate the total surface area:
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Calculate the area of both circular bases: \[ 2 \cdot 452.16 = 904.32 \text{ cm}² \]
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Add the lateral surface area: \[ \text{Total Surface Area} = 904.32 \text{ cm}² + 678.24 \text{ cm}² = 1582.56 \text{ cm}² \]
Thus, the surface area of the steamer basket is:
\[ \boxed{1582.56 \text{ cm}²} \]