This problem involves a right triangular prism. Let's break it down step by step.

Given:

• The cross-sectional area of the right triangular prism is a right triangle with sides of 6 cm, 8 cm, and 10 cm (this is a Pythagorean triplet).
• The surface area of the original prism is 264 cm².
• The goal is to find the height of the original prism.

Step 1: Find the area of the triangle.

Since the triangle is a right triangle, the area $A$ of the triangle can be calculated using the formula for the area of a right triangle: $A = \frac{1}{2} \times \text{base} \times \text{height}$ Here, we can take the base as 6 cm and the height as 8 cm (the perpendicular sides of the right triangle). So, $A = \frac{1}{2} \times 6 \times 8 = 24 \, \text{cm}^2$

Step 2: Use the surface area formula.

The surface area $S$ of the prism is given by the formula: $S = 2 \times A + \text{perimeter of base} \times \text{height of the prism}$ Where $A$ is the area of the triangular cross-section, and the perimeter of the base triangle is the sum of the sides (6 cm, 8 cm, and 10 cm): $\text{Perimeter} = 6 + 8 + 10 = 24 \, \text{cm}$ The surface area $S$ is given as 264 cm², so: $264 = 2 \times 24 + 24 \times h$ Where $h$ is the height of the prism.

Step 3: Solve for $h$.

Simplifying the equation: $264 = 48 + 24h$ Subtract 48 from both sides: $216 = 24h$ Now, solve for $h$: $h = \frac{216}{24} = 9 \, \text{cm}$

Final Answer:

The height of the original prism is 9 cm.

Would you like a more detailed explanation, or do you have any questions? Here are some related questions you can explore:

1. How do you calculate the surface area of other types of prisms?
2. What happens if the triangle is not a right triangle?
3. How do you calculate the volume of a triangular prism?
4. What is the relationship between the surface area and the height of a prism?
5. Can the height of a prism be found if only the base area and the lateral surface area are given?

Tip: Always double-check the geometry of the base when calculating surface areas of prisms, as it can significantly affect the formula used!