To find the surface area of the right triangular prism, we need to calculate the areas of its two triangular bases and its three rectangular faces.

Step 1: Find the area of the triangular base

The triangular base has a base of 12 inches and a height of 15 inches.

$\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times 15 = 90 \, \text{in}^2$

Since there are two triangular bases, the total area of the two triangles is:

$2 \times 90 = 180 \, \text{in}^2$

Step 2: Find the areas of the three rectangular faces

The prism has three rectangular faces:

1. Face 1: The rectangle formed by the prism's height (9 inches) and the triangular base's side (15 inches). $\text{Area of Face 1} = 9 \times 15 = 135 \, \text{in}^2$

2. Face 2: The rectangle formed by the prism's height (9 inches) and the base of the triangle (12 inches). $\text{Area of Face 2} = 9 \times 12 = 108 \, \text{in}^2$

3. Face 3: The rectangle formed by the prism's height (9 inches) and the hypotenuse of the triangular base. First, we need to calculate the hypotenuse using the Pythagorean theorem:

   $\text{Hypotenuse} = \sqrt{12^2 + 15^2} = \sqrt{144 + 225} = \sqrt{369} \approx 19.21 \, \text{in}$ Now, calculate the area of Face 3: $\text{Area of Face 3} = 9 \times 19.21 \approx 172.89 \, \text{in}^2$

Step 3: Calculate the total surface area

The total surface area is the sum of the areas of the two triangles and the three rectangular faces:

$\text{Total Surface Area} = 180 + 135 + 108 + 172.89 \approx 595.89 \, \text{in}^2$

Thus, the surface area of the triangular prism is approximately 595.89 square inches.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you calculate the volume of a triangular prism?
2. What is the lateral surface area of a triangular prism?
3. How do the dimensions of the triangular base affect the surface area of a prism?
4. What is the importance of the Pythagorean theorem in 3D geometry problems?
5. How would the surface area change if the height of the prism increased?

Tip: In surface area calculations, always carefully account for each unique face of the shape.