Isabella is making sushi. She slices a piece of salmon into the shape of a solid triangular prism. The prism is $2$ cm thick, and its triangular faces have side lengths $7$ cm, $ 24$cm, and $25$ cm. Find the volume of this piece of salmon in cm$^3$.

[i]Proposed by Isabella Li[/i]
1. **Identify the shape and dimensions**: The problem states that the solid is a triangular prism with a thickness (height) of \(2\) cm. The triangular base has side lengths \(7\) cm, \(24\) cm, and \(25\) cm.

2. **Verify the triangle**: Check if the triangle with sides \(7\) cm, \(24\) cm, and \(25\) cm is a right triangle. We use the Pythagorean theorem:
   \[
   7^2 + 24^2 = 49 + 576 = 625 = 25^2
   \]
   Since \(7^2 + 24^2 = 25^2\), the triangle is indeed a right triangle.

3. **Calculate the area of the triangular base**: For a right triangle, the area \(B\) can be calculated using the formula:
   \[
   B = \frac{1}{2} \times \text{base} \times \text{height}
   \]
   Here, the base and height are \(7\) cm and \(24\) cm respectively:
   \[
   B = \frac{1}{2} \times 7 \times 24 = \frac{1}{2} \times 168 = 84 \text{ cm}^2
   \]

4. **Calculate the volume of the prism**: The volume \(V\) of a prism is given by:
   \[
   V = B \times h
   \]
   where \(B\) is the area of the base and \(h\) is the height (thickness) of the prism. Given \(B = 84 \text{ cm}^2\) and \(h = 2 \text{ cm}\):
   \[
   V = 84 \times 2 = 168 \text{ cm}^3
   \]

The final answer is \(\boxed{168}\) cm\(^3\).