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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.

A right triangular prism is shown below. What is the surface area of the triangular prism?

Learn how to calculate the surface area of a right triangular prism with a triangular base of 12 inches and height 9 inches. This problem involves using geometry concepts and the Pythagorean theorem. The solution covers step-by-step calculations for the triangular base and rectangular faces. Suitable for Grades 7-10.

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