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The problem shows a triangular prism with a given volume of 198 cm³, where the cross-section is a right-angled triangle. The dimensions of the triangle are provided as 4 cm (one leg) and \(x\) cm (the other leg), with the length of the prism being 9 cm.

We are asked to calculate the value of \(x\).

### Step-by-step solution:

1. **Volume of a triangular prism formula**:
   The volume \(V\) of a triangular prism is given by:
   \[
   V = \text{Area of cross-sectional triangle} \times \text{length of prism}
   \]
   
2. **Area of the right-angled triangle**:
   Since the cross-section is a right-angled triangle, the area \(A\) of the triangle can be calculated using the formula for the area of a triangle:
   \[
   A = \frac{1}{2} \times \text{base} \times \text{height}
   \]
   Here, the base is 4 cm and the height is \(x\) cm, so:
   \[
   A = \frac{1}{2} \times 4 \times x = 2x \, \text{cm}^2
   \]

3. **Volume equation**:
   The total volume of the prism is given as 198 cm³, and the length of the prism is 9 cm. Using the formula for the volume of a triangular prism:
   \[
   V = \text{Area} \times \text{length}
   \]
   Substituting in the known values:
   \[
   198 = (2x) \times 9
   \]
   Simplifying:
   \[
   198 = 18x
   \]
   
4. **Solve for \(x\)**:
   \[
   x = \frac{198}{18} = 11 \, \text{cm}
   \]

So, the value of \(x\) is **11 cm**.

Would you like more details or have any further questions?

Here are 5 related questions for further practice:
1. What is the formula for the surface area of a triangular prism?
2. How would the volume change if the length of the prism doubled?
3. What would the height of the triangle be if the volume were 396 cm³ and the other dimensions stayed the same?
4. How do you calculate the lateral surface area of this prism?
5. What is the volume of a prism with a cross-sectional area of 30 cm² and a length of 12 cm?

**Tip:** Always remember to double-check units when calculating volume or area, to ensure consistency throughout the solution!

The diagram shows a triangular prism. The cross-section of the prism is a right-angled triangle. The volume of the prism is 198 cm³. Calculate the value of x.

Calculate the value of x in a triangular prism where the volume is 198 cm³, the length is 9 cm, and the cross-sectional triangle has a base of 4 cm. This problem covers key geometry concepts like calculating the volume of a prism and the area of a right-angled triangle.

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